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11. The sine, tangent, &c., of the arc BC, are often called the sine, tangent, &c., of the angle BOC, to the radius OC.
12. Sometimes also they are called the sine, tangent, &c., of the number of degrees contained in the arc BC or in the angle BOC, the radius being at the same time named or understood. Thus, if the arc BC, or the angle BOC, is of 50°, and the radius OC is 10 of any equal parts, then BD is called the sine of 50°, CF the tangent of 50°, &c., to the radius 10.
. EXERCISE 1. If the radius is 10, what is the sine of 90°; what is the chord of 60°; what, the sine of 30°; what, the cosine of 60°; and what, the versed-sine of 60° ? 2. If the radius is 1, what is the tangent of 45° ; what, the cotangent of 45°; what, the secant of 60°; what, the cosine of 0°; and what, the tangent of 90° ? 3. The radius being 1, compute the sine and tangent of 60°, and the secant of 45°.*
13. When the arc BC is greater i than a quadrant, the sine BD, the versed-sine DC, the tangent CF, the secant OF, the cosine BH or OD, the co-versed-sine HG, the cotangent GI, and the cosecant OI, occupy the places shown in the annexed diagram.
COR. The sine of an arc is equal to the sine of its supplement.
EXERCISE. What is the sine of 30°; what, of 150°; what, of 60°; and what, of 120°; the radius being 100 ?
14. For use in calculation, the sines, tangents, secants, and versed-sines, of arcs, for every minute in the quadrant, to the radius 1, have been computed and arranged in tables; and, on the discovery of logarithmic arithmetic, the logarithms of the sines, tangents, secants, and versed
• The scholar is supposed to be acquainted with the first principles of Practical Geometry and of Mensuration, and nothing more is required for the above Exercises. The answers, when involving decimal fractions, may be carried out as far as the teacher pleases.
sines, for every minute of the quadrant, to the radius 10,000,000,000, were arranged in similar tables. The latter are called Logarithmic Sines, Tangents, &c. : they may be found in Table III of this volume. In contradistinction to these, the former are called Natural Sines, Tangents, &c.; but such is the facility which logarithms afford, that the natural sines, &c., have now been almost superseded by the logarithmic, and are seldom used.
15. The words sine, tangent, secant, versed-sine, cosine, cotangent, cosecant, and co-versed-sine, are often abbreviated thus : sin, tan, sec, versin, cos, cot, cosec, and coversin.
ON THE USE OF THE TABLES OF SINES, TANGENTS, AND SECANTS.*
To find, from Table III, thé Logarithmic Sine, Tangent, Secant, Cosine, Cotangent, or Cosecant of an acute Angle † of any given number of Degrees and Minutes.
RULE. If the given number of degrees is less than 45, look for the degrees in the top line of the table, turning over the leaves till the proper page is found. In that page, look in the second line for the name of the column wanted; and on the left margin of the page for the given number of minutes. But, if the given number of degrees is not less than 45, look for the degrees in the bottom line of the page; for the name of the column, in the second line from the bottom ; and for the minutes, on the right margin. In either case, having found the minutes, then, in the same line, in its proper column, will be found the logarithm wanted.
* These, of course, also include cosines, cotangents, and cosecants, which are merely sines, tangents, and secants of the complemental arcs or angles. † Or of an arc less than a quadrant. But it is properly angles and not arcs on which Trigonometry is employed. Some given radius is understood, which is, of course, the radius of the table used.
EXERCISE 1. What is the logarithmic sine of 9° 10' ? Ans. 9.202234. 2. What is the log. tangent of 37° 26'? Ans. 9.883934. 3. Find the log. secant of 64° 19'. Ans. 10:363114. 4. Find the log. cosine of 88° 44'. Ans. 8:344504.
To find, from Table III, the Logarithmic Sine, Tangent, or Secant of an Acute Angle of any given number of Degrees, Minutes, and Parts of a Minute.
RULE. Neglect the parts of a minute; but, if they amount to more than half a minute, increase the number of minutes by a unit. Then proceed as in the last problem.
EXAMPLE 1. What is the logarithmic sine of 22° 32' 48" ?
Log sine of 22° 33'............9-583754, Ans.
EXAMPLE 2. What is the log tangent of 28° 47' ? Log tan 28° 48'............9.740169, Ans.
EXAMPLE 3. Find the log. secant of 56° 22:38'. Log sec 56° 22' ......... .10.256587, Ans.
COR. The logarithmic cosine, cotangent, and cosecant may be found in the same manner as the sine, tangent, and secant.
EXAMPLE 4. Find the logarithmic cosine of 26° 13.55'. Log cos 26° 14'............9.952793, Ans.
EXERCISE. Find the logarithmic tangent of 33° 17' 57" ; the log. sine of 68° 153'; and the log. secant of 49° 28.84'. Answers: 9.817484; 9.967927; and 10.187308.
To find the Logarithmic Sine, Tangent, Secant, Cosine, Cotan gent or Cosecant of a given Obtuse Angle.
EXAMPLE. What is the logarithmic sine of 129° 37' ? 180° 0 129° 37'
Log sine of 50° 23'............9.886676, Ans.
EXERCISE. What are the logarithmic sines of 170° 50', and of 91° 16' ? Answers: 9.202234, and 9.999894.
To find, from Table III, the Number of Degrees and Minutes, corresponding to any given Logarithmic Sine, Tangent, or Secant.
RULE. Look for the given sine, tangent, or secant, in its proper column; and, having found it exactly (or, if not exactly, the nearest to it whether larger or smaller), the minutes will be found on the same line on either the right or the left margin, and the degrees at the top or bottom of the page. That is, if the name, “sine,” “ tangent,” or secant," is found at the head of the column, then the degrees are to be taken from the top of the page, and the minutes from the left margin: but, if the name is found at the foot of the column, then the degrees are to be taken from the bottom of the page, and the minutes from the right margin.
Cor. The degrees and minutes corresponding to a given logarithmic Cosine, Cotangent, or Cosecant, are found in the
NOTE. It will depend upon the nature of the question whether the angle required is that found from the table or its supplement, except in the case of the versed-sine.
EXERCISE 1. What is the number of degrees and minutes corresponding to each of the logarithmic sines, 9.731009; 9.871073; 9.556978; and 9.943939 ? Answers: 32° 34'; 48°; 21° 8'; and 61° 31'. 2. What are the degrees and minutes answering to the log. tangent 10.047850; to the log. tangent 9.787000; and to the log. secant 10.043700 ? Answers : 48° 9'; 31° 29'; and 25° 16'. 3. What are the degrees and minutes answering to the log. cosine 9.862386; log. cotangent 9.787954; and log. cosecant 10:307796 ? Answers : 43° 15'; 58° 28'; and 29° 29'.
OF RIGHT-ANGLED TRIANGLES.
Throughout this chapter, b represents the base; p, the perpendicular; h, the hypothenuse; and B, P, and H, the angles respectively opposite the sides b, p, and h.
In a right-angled Triangle, having given one of the two acute Angles, to find the other.
RULE. Subtract the given angle from 90°. (See Def. 6 in Ch. v.)
EXERCISE 1. One of the acute angles of a right-angled triangle is 371°: the other is required. Ans. 52) 2. What is the acute angle at the vertex when that at the base is 79° 48' ? Ans. 10° 12'. 3. What is the acute angle at the base when that at the vertex is 54° 28.385' ? Ans. 35° 31.615'. Page 3
4. The acute angle at the base being 11° 54' 8" 38"', what is the angle opposite the base ? Ans. 78° 5' 51" 22".
Having given the Hypothenuse and the Angles of a right-angled Triangle, to find the Legs.
RULE. As the radius * is to the sine of one of the acute angles, so is the hypothenuse to the leg opposite the said angle.
NOTE 1. Instead of the sine of the angle opposite the required side, we may use the cosine of the acute angle adjacent to it, when the latter angle, and not the former, is given : it will save the trouble of subtracting. But it will be perceived that, in the table, the two are identical, the subtraction being performed, when there are no seconds, by merely looking from the left margin to the right for the minutes, and from the top of the page to the bottom for the degrees.
NOTE 2. It will be useful to the scholar, throughout all the exercises in Trigonometry, to draw the diagrams, marking the given sides and angles with their numerical values. It will also furnish useful exercises of Practical Geometry to draw such diagrams correctly, from a scale, and to ascertain from the same scale the required parts. The geometrical process should be a proof of the arithmetical. The diagram to the next example will illustrate this; but in future it will be left to the student to draw and mark the figure for himself.
* By this is meant the radius used in the tables employed. If the table of natural sines is used, then the radius is 1; if the logarithmic table, the radius is 10,000,000,000, and its logarithm is 10. + One formula is sufficient, since either leg may be made the base; but the double mode of expression will be clearer to beginners.
EXAMPLE 1. The hypothenuse of a right-angled triangle is 63, and one of the acute angles 30°. What is the side opposite that angle? By Chap. 11, Problem XII, As radius...... 10.000000
To 1.498311. Opposite this logarithm, in the column of sines in Table II, we find 31.5 as before.
EXAMPLE 2. The acute angle at the base is 18° 9' 42", and the hypothenuse is 0.69848. The base is required by logarithms. As radius.... .10.000000
Is to cos 18° 10'.......9.977794 (See Note 1, page 37). So is 6.985... .0•844166 (See Note 3, page 16).
To 6.637 – .....0.821960 Ans. 0.6637 - .
EXERCISE 1. The hypothenuse of a right-angled triangle is 10:47, and the acute angle at the base is 58° 20'. Compute the length of the perpendicular. Ans. 8.91+. 2. The base of a right-angled triangle is required, the hypothenuse measuring 89 yds. 2 ft., and the acute angle at the vertex 35°. Ans. 51 yds. 1 ft. +. 3. The acute angle at the base is 671°, and the hypothenuse 4.913: what is the base ? Ans. 1.900. 4. The diagonal of a rectangular field is 344 links, making an angle of 22° 14' with the longer side: what are the sides ? Ans. 130.2 -- and 318.4 +. 5. The hypothenuse is 0.396,* and one of the acute angles 15° 28': what is the opposite side ? Ans. 0.1056 +. 6. The hypothenuse is 98·324,f and one of the acute angles is 48° 15•8': $ calculate the opposite side. Ans. 73.4-.
* See Chapter 11, Problem XII, Note 3. † See Chapter 11, Problem vi. See Chapter vi, Problem II.
7. The hypothenuse is 0.48285,* and one of the acute angles 39° 42' 15":f what are the legs? Answers : 0•308 +, and 0.371 +.
In a right-angled Triangle, having given the Angles and one Leg, to find the Hypothenuse.
RULE. As radius is to the secant of the acute angle adjacent to the given side, so is the given side to the hypothenuse.
NOTE. Instead of the secant of either of the acute angles, we may, if we please, use the cosecant of the other acute angle. (See Note 1 to Problem 11.)
EXAMPLE. The base of a right-angled triangle is 29:42, and the acute angle at the vertex is 58° 30': what is the hypothenuse? ZP=compl. of Z B = 31° 30'. As radius........ .10.000000
Is to sec (P=31° 30') .......10.069234 1.468643
EXERCISE 1. The base of a right-angled triangle is 315, and the adjacent acute angle is 60°: the hypothenuse is required. Ans. 63. 2. What is the hypothenuse when the perpendicular is 1900, and the vertical angle 67° 15' ? Ans. 4913 +. 3. The base is 272.8, and the acute angle at the vertex 60°: what is the hypothenuse ? Ans. 315, very nearly.
* See Chapter 11, Problem vr.
+ See Chapter vi, Problem i.
4. The perpendicular is 78-324, and the acute angle at the base 28° 49' 45" : what is the hypothenuse ? Ans. 162.4+. 5. The base is 0.076, and the vertical angle 14° 37°12': what is the hypothenuse? Ans. 0•301 +.
In a right-angled Triangle, having given the Angles and one Leg, to find the other Leg.
RULE. As the radius is to the tangent of the acute angle adjacent to the given leg, so is the given leg to the other leg.
Radius : tan P ::6: p.
(See the diagram on the following page.)
The second formula is the same as the first, applied to the other angle.
NOTE. Instead of the tangent of the acute angle adjacent to the given leg, we may take the cotangent of the angle opposite that leg.
EXAMPLE. The base of a right-angled triangle is 31.5, and the acute angle at the base 60°: the perpendicular is required.
EXERCISE 1. The base of a right-angled triangle is 69:4; and the acute angle at the base, 45°: what is the perpendicular ? Ans. 69.4. 2. The perpendicular is 4,984, and the vertical angle is 33° 17' : what is the base ? Ans. 3272 -.
3. The acute angles of a right-angled triangle are 35 and 55 degrees. The leg opposite the latter is 9 feet 5 inches: the other leg is required. Ans. 6 ft. 7 in. 4. Base, 94:738; adjacent angle, 48° 23' 10": find the perpendicular. Ans. 106:6 +. 5. Perpendicular, 0:106654; vertical angle, 41° 3683': find the base. Ans. •0947 +.
In a right-angled Triangle, having given the Hypothenuse and one Leg, to find the Angles.
RULE 1. As the hypothenuse is to the given leg, so is the radius to the sine of the angle opposite the given leg.
RULE 2. The one leg is to the hypothenuse, as the radius to the secant of the angle adjacent to the said leg.
NOTE. Having found the one angle, the other is found by Problem 1. Or, if we choose, we get the other at once, by taking the fourth term of the proportion as a cosine or a cosecant instead of a sine or a secant.
EXAMPLE. The hypothenuse is 63, and the perpendicular 31.5: the angles are required. As 63 ...... 1•799341
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As 31.5 ............ 1.498311
Is to 63 ............. 1•799341
To sec 60° .........10301030. Ans. ZB= 60° ..ZP= 30°.
EXERCISE 1. The hypothenuse is 10:47, and the perpendicular 8.91: find the acute angle at the base. Ans. 58° 19' +. 2. What is the angle at the vertex, the hypothenuse being 89 yds. 2 ft., and the base 51 yds. 1 ft. ? Ans. 34° 55' +. 3. The base is 1900, and the hypothenuse 4913: what is the acute angle at the base ? Ans. 67° 15' -. 4. The hypothenuse being 5.235, and the perpendicular 4.455, the vertical angle is required. Ans. 31° 41'-. 5. Hypothenuse, 034; base, .028: the angles are wanted. Answers : 34° 34' - , and 55° 26' +. 6. Hypothenuse, 16,386; perpendicular, 9,569 : find the angles. Answers : 54° 17' -, and 35° 43+.
In a right-angled Triangle, having given the Legs, to find the acute Angles.
RULE. As either leg is to the other leg, so is radius to the tangent of the angle opposite the latter.
So:p :: radius : tan P. p :b :: radius : tan B.
(See the diagram on the preceding page.)
EXAMPLE. The base is 3, and the perpendicular 4: what are the angles ?
may either say, As 3 : 4 :: radius : tan P, or As 4 : 3 :: radius : tan B. :
We take the former. As 3 ...
Is to 4 ...........
To tan 53° 8'............10.124939. Ans. Z P=53° 8' .. ZB = 36° 52'.
EXERCISE 1. The two legs are 31.50 and 54:56: the angles are required. Ans. 30° and 60°, nearly. 2. The base is 6.94, and the perpendicular the same : what are the angles ? Ans. Each 45°. 3. The perpendicular is 4,984, and the base, 3,272: what is the vertical angle ? Ans. 33° 17' +. 4. The perpendicular is 9 ft. 5 in., and the base 6 ft. 7 in. : what is the acute angle at the base. Ans. 55°, nearly. 5. Given, the legs .07859 and .06990: required the angles. Ans. 41° 39' + and 48° 21' -. 6. The legs are 20,671 and 9,473.8: the angles are wanted. Ans. 24° 37' + and 65° 23'
If it be asked how the preceding rules are to be kept in memory without reference to the book, observe the following direction : Suppose a circle to be described with a given side (whether leg or hypothenuse) as radius. Then observe what the other given or required side becomes with reference to the circle just described,—that is, whether the sine, tangent, or secant; and write that on the last-mentioned side as its name. The name of that side will thus be the sine, tangent, or secant, of the opposite angle. This being done, if one of the three sides is required, say :-As the radius of the table, to the name written on the other side, so is the side used as radius, to the other side. But, if an angle is required, say :-As the side used as a radius, is to the other given side, so is the radius of the table to the name written on the other side.
This mode of proceeding is what is usually called “Making a given side Radius."
In a right-angled Triangle, having given the Hypothenuse and one Leg, to find the other Leg.
Rule 1. Subtract the square of the given leg from the square of the hypothenuse, and extract the square root of the remainder.
For EXAMPLES of the application of this rule and of Rule 1 of the next problem, as well as for numerous ExERCISES in both, see Part II, Ch. III, Pr. I. The scholar may also, if he choose, work the following exercises by this rule.
RULE 2. Multiply the sum of the given leg and hypothenuse by their difference, and extract the square root of the product. Sb=^{(h+p) x (h-p)}. FORMULÆ. p=w{(h + b) x (h-6)}.
EXAMPLE 1. The hypothenuse is 654 and the perpendicular 583: what is the base ? 654 583 Logarithms. Sum =1237 ..3.092370 Difference 71. .1.851258
296.4 - ............2471814.
RULE 3. Find the acute angles by Problem v, and then the required leg by Problem 11.
NOTE 1. It is not necessary to find the angles themselves, but merely their sines (natural or logarithmic): for, having obtained the sine of the one acute angle, and found the nearest to it on the table, we find the sine of the other acute angle (when great accuracy is not required) by merely looking along the line from the right hand column of sines to the left, or from the left to the right. NOTE 2. After finding the angles, we may, if we choose, use Problem iv, instead of Problem II, for finding the required leg
EXAMPLE 2. Find the answer to Example 1 by Rule III. As 654.......... 2.815578 | As radius.......10.000000
Is to 583....... 2:765669 | Is to sin <B... 9•656302 So is radius,...10.000000 | So is 654 ....... 2·815578
To sin Z P..... 9.950091 To b= 296.4 + 2.471880 Ans. 296.4+.
EXERCISES IN RULES 2 AND 3. 1. The hypothenuse is 57.97, and the perpendicular 51.15: compute the base. Ans. 27.3 -. 2. The hypothenuse is 617.2, and the base 493.8: compute the perpendicular. Ans. 370 +. 3. The hypothenuse is 2.4286, and the one leg 2.1429 : what is the other leg ? Ans. 1.143. 4. The hypothenuse being 0.126731, and the one leg ·027819, what is the other leg ? Ans. 0.124-.
In a right-angled Triangle, having given the two Legs, to find the Hypothenuse.
RULE 1. Add together the squares of the two legs, and extract the square root of the sum. FORMULA. h=N(+ po).
RULE 2. Find either acute angle by Problem vi, and then the hypothenuse by Problem III.
NOTE. See Note 1 to the last Problem.
EXAMPLE. The base is 0.24915, and the perpendicular 0:59796 : what is the hypothenuse ?
As radius.......10.000000
To 5.980.......0•776701 To sec P........10:415032 So is radius...10.000000 So is 2.491..... 0•396374
To tan P......10:380327. To 6.477 + .... 0·811406.
EXERCISES IN RULE 2. 1. The base is 27.28, and the perpendicular 51.15 : determine the hypothenuse. Ans. 58, very nearly. 2. The two legs are 493.76 and 370:32 : the hypothenuse is required. Ans. 617 +. 3. The legs are 1.1429 and 2:1429 : find the hypothe Ans. 2.429. 4. What is the hypothenuse if the legs are 0·12364 and .027819 ? Ans. 0·127
PROMISCUOUS EXERCISES IN RIGHT-ANGLED TRIANGLES.
1. The base is 6; and the acute angle at the base, 67° : what is the hypothenuse, and the perpendicular? Ans. 15.85 –, and 14:67 – 2. The hypothenuse is 3,965; and the base, 3,172: what is the perpendicular ? Ans. 2,379. 3. The hypothenuse is 100, and one of the acute angles four times the other : what are the sides? Ans. 30.90 +, and 95.11-. 4. The one leg of a right-angled triangle is four times the length of the other : what are the acute angles ? Ans. 75° 58' —, and 14° 2' +. 5. Each of the sides of an isosceles triangle is four times its base : what are its angles ? Ans. 14° 22' –, 82° 49' + , and 82° 49' +. 6. The diameter of a circle is 1,000 : what is the chord of an arc of 30° ? Ans. 258.8 +. Page 5
EXERCISE 1. Two angles of a triangle are 58° 12', and 64° 33' ; the side opposite the former is 385 : what is the side opposite the latter ? Ans. 409 +. 2. The three angles of a triangle are 38° 24', 46° 31', and 95° 5'; the greatest side is 7.832 : what is the least side ? (See the Note). Ans. 4.884 +. 3. In the triangle described in the last exercise, what is the third side ? Ans. 5•705 +. 4. Two of the angles of a triangular field being 63° 55' and 48° 27', and the side opposite the latter 54 chains: what are the other two sides ? Ans. 6.901 –, and 7.105 + chains. 5. Each of the equal angles at the base of an isosceles triangle is 61° 47', and the base 39 ft. 5 in.: what are the sides ? Ans. Each 41 ft. 8 in. +. 6. Two of the angles of a triangle are 35° 12', and 116° 33'; the side opposite the former is 0.5876 : what are the other two sides? Ans. 0.9119 –, and 0.4825 — . 7. One side of a triangle measures 784:36 feet, and the opposite angle 64° 31.25' : what is the side opposite a second angle of 91° 4.75' ? Ans. 869 – feet.
In any Triangle, having given two Sides and an Angle opposite one of them, to find the other Angles.
RULE. As the side opposite the given angle is to the other given side, so is the sine of the given angle to the sine of the angle opposite the latter side. This being found, the third angle is obtained by Prob
A, C, a, and c, being the same as in the last problem.
The rules for both problems are expressed by the following THEOREM :—The Sides of any Plane Triangle are to each other as the Sines of the opposite Angles.
NOTE. Since, by the rule, we find the sine of the required angle, and not the angle itself, and since the sine of any angle and that of its supplement are the same, we always obtain two values for the angle found by the rule, -one less than 90°, the other greater : some directions are therefore necessary, to know which of the two values should be taken. Page 6
3. The greatest side of a triangle is 7.832, and the least side 4.884 ; the greatest angle is 95° 15' : compute the third side. Ans. 5.69 +. 4. Two sides of a triangle are 3 and 4; the angle opposite the latter is 78° 14': what is the third side ? Ans. 3:33 5. Two sides of a triangle being •069014 and .057498, and the angle opposite the latter being 48° 27•2', what is the remaining side ? Ans. •0711-, or ·0205 –
In any Triangle, having given two Sides and the included Angle, to find the other Angles.
RULE. Subtract the given angle from 180° : the remainder will be the sum of the two required angles. Then say,
As the sum of the two given sides is to their difference, so is the tangent of half the sum of the two required angles, to the tangent of half their difference.
Having found the said half difference, add it to the half sum of the same two angles (previously found) for the greater angle, or subtract it from the half sum for the smaller angle.
a+c: a-c:: tan ) (A + C) : tan } (A - C).
A= (A + C) + (A-C). = } –
a and c being the given sides, B the given angle, and A and C the required angles.
Two sides of a triangle (a and c) are 794:6 and 724:4 respectively; and the angle (B) contained between them is 36° 46' : what are the other angles ? 180° 0'
794:6=a 36 46= 2B 724.4=c
EXERCISE 1. Two sides of a triangle are 409 and 381, and the angle included between them is 58° 12' : what are the other angles ? Answers : 64° 33' – , and 57° 15' +. 2. The vertical angle of a triangle is 23° 46', and the two sides 9.72 and 8:88: what are the two angles at the base ? Answers : 90° 14' –, and 66° 0' +. 3. One angle of a triangle is 97° 26', and the two sides which contain it are 0·843 and 0.657: what are the angles opposite those sides ? Answers : 47° 30' –, and 35° 4' +. 4. Two sides of a triangle are 7,103.5 and 6,901:4; their included angle is 48° 27'2' : what are the other angles Answers : 67° 36' +, and 63° 56' +.
In any Triangle, having given two Sides and their included Angle, to find the third Side.
RULE. Find the other angles by Problem y, and then the third side by Problem II.
NOTE. After finding the two unknown angles, only one of these and one of the two given sides is employed to find
the third side. Of the said two angles, the greater should always be taken in preference to the smaller, and consequently, also, the greater of the two given sides. These give the more accurate result. But if the said angles and sides are not very unequal, the smaller may be used for a second computation in proof of the first. The true answer will always be between the two results, but nearest to that obtained by using the greater side and angle.
EXERCISE 1. Two sides of a triangle are 8.18 and 7.62, and their included angle is 58° 12' : what is the third side ? Ans. 7.70 -. 2. The vertical angle of a triangle is 47° 32', and the two sides 57.3 and 48.4: what is the base ? Ans. 43:37. 3. Two sides of a triangle being 0·12 and 0.05, and including an angle of 90 degrees : compute the third side by this rule. Ans. 0:13. 4. Two sides of a parallelogram are 39•7 and 48:3, and each of its acute angles is 56° 14': what are the diagonals ? Answers : 77.7 +, and 42.2 5. Find the base of a triangle which has its two sides 71.035 and 69·014, and the vertical angle 48° 27•2'. Ans. 57.5 -.
In any Triangle, having given the three Sides, to find the Angles.
RULE. In an acute-angled triangle call any one of the three sides the base ; but in an obtuse-angled triangle call the longest side the base. Draw a perpendicular, BD, from the vertex to the base, or suppose it drawn. Then say, As the base is to the sum of the two sides, so is the difference of the two sides, to the difference of the segments of the base. Half the difference, thus found, added to half the base, will give the greater segment, CD; and half the difference, subtracted from P half the base, will give the smaller segment, AD. Having found these, the angles
C and A are found, by Problem v of the last Chapter ; and then the angle B, by Problem 1 of this Chapter.
6:a +0 :: a-C: d. CD=6+ d. AD=- £d. CD : a :: radius : sec C. AD : 0 :: radius : sec A. d being the difference of the segments of the base.
EXAMPLE. The three sides of a triangle are 28, 40, and 16: what are the angles ?
NOTE. The rule proceeds on the supposition that the two sides are unequal. If they are equal, the first part of the rule is evidently not required, since, in that case, the segments are each half the base.
EXERCISE 1. The three sides are 735, 595, and 350 feet : what are the three angles ? Ans. 28° 4' +, 98° 48', and 53° 8'2. What are the three angles, supposing the three sides respectively 0.532, 0.259, and 0.371 miles ? Ans. 39° 33' +, 26° 24' –, and 114° 3' 3. The base of a triangle measures 64 ft. 87 in., and the two sides respectively 45 ft. 7} in., and 36 ft. 3 in. : what are the angles ? Ans. 32° 57' +, 43° 12' +, and 103° 51' -.
OF THE APPLICATIONS OF PLANE TRIGONOMETRY.
The principal applications of Trigonometry are to the measurement of Heights and Distances by means of angular observations, to Navigation, to Astronomy, and to Mechanics and the other departments of Natural Philosophy. Trigonometry is much employed in Land-surveying, when conducted on an extensive scale, such as the National Survey of the British Isles, but the processes employed are the same as those for the determination of Heights and Distances. When a survey is spread over so large a portion of the Earth's surface that it cannot be regarded as a plane, the aid of Spherical Trigonometry then becomes essential. Questions relating to that class of operations do not belong to the subject of this volume. Many works on Trigonometry describe various simple modes of ascertaining unknown heights and distances entirely independent of angular measurements, and include questions relating to these methods. Such questions belong more properly to Mensuration of Lines. In the course of the following Exercises, allusion will frequently be made to instruments employed for the mea Page 7
the index is at zero. In adjusting the instrument previously to the observation, the index is fixed at zero, the telescope is turned towards the object, and the stem is brought as nearly as possible to an upright position by means of the feet on which it stands; and, when that is done, a screw acting on the stem brings the spirit level to a horizontal position and completes the adjustment in the direction of the telescope, while that in the cross direction, being considered of comparatively little importance, is determined by the eye, and attained by the use of the feet of the instrument only. The exact point to which the telescope is directed is determined by two hairs or other fine lines, crossing the sight at right angles to each other, -one being horizontal, the other vertical. The quadrant in either of the preceding forms is used only for vertical angles, that is, angles in a vertical plane. But even for that purpose it is an imperfect instrument, and is not employed in nice operations. In these it is superseded by the following: The Theodolite consists of a vertical semicircle, S, connected with a horizontal circle, C, both graduated at the circumference. The graduated horizontal circle remains fixed during an observation, while another concentric circular plate, I, turns upon it, having an index on its margin: but the vertical arc moves, while its index remains fixed. While the vertical semicircle and the horizontal index-circle turn freely, each on its own axis, the axes themselves are firmly connected with each other. Two spirit levels, L', L', placed the circle, at right angles to each other, determine its accurate position in a horizontal plane, and, at the same time, the position of the semicircle in a vertical plane; the instrument being brought nearly into that Page 8
directly opposite, 41° 15'. Then, guided by a plumb line, I mark two other points on the wall directly beneath that observed at the top,—the one at the base,—the other 5 feet higher, corresponding to the height of the quadrant,-and find the elevation of the latter 18° 52'. I then suspend the plummet from the quadrant to the ground, and ascertain the distance from the plummet to the mark at the base of the wall, measured on the slope, to be 384 feet. What is the height of the wall from its base ? Ans. 199.5 – feet. 18. Placing a quadrant on a plane but not level ground, 90 feet from the base of an upright corner of a building, the distance being measured on the sloping ground, as described in Exercise 17, I take the angle of the summit of the corner, and find it 39° 27'. I then bring the telescope of the quadrant to a horizontal position, turned towards the corner, and, marking the point to which it is now directed, I find the point thus marked to be 26 feet above the base, the height of the quadrant being 5 feet. What is the height of the wall ? Ans. 98:01 + feet. 19. From the brow of a steep headland a ship was observed in the roads beneath. The angle of depression of the ship, at the water line, was found to be 9° 28'; and that of the shore immediately below, in the direction of the ship, 72° 40'. What was the distance of the ship from the point of observation, and what from the shore, the perpendicular height of the precipice being known to be 254 feet, and the surface of the sea being regarded as a plane ? Ans. 1,544 and 1,444 feet, nearly. 20. Two streets meet at an acute angle. The one lies N 51° W, and the other S 48° W. The distance from the corner to a druggist's shop door in the first street is 315 yards; and the distance from the same corner to a surgeon's door in the other street is 406 yards. What is the distance in a straight line from the surgeon's door to the druggist's ? Ans. 473 + yards. 21. From a vessel at anchor two rocks are observed to the westward, the one (A) bearing WNW; and the other (B), W by S, from the vessel. The rocks, being well known, are laid down on the chart, from which it is found that the former bears NNE from the latter, distant 645 yards. What are their respective distances from the vessel ? Ans. A is distant 965, and B, 1,161 yards, nearly. 24. A castle-wall rises perpendicularly from the shore of a lake, its base being washed by the water. A rock Page 9
telescope of the theodolite at E and F is required, in feet, taking the surface of the earth as a plane (that is, the line PR); and also the horizontal distance of the mountain top from the station E, in miles (that is, the line ER). Ans. ER=8.744 miles, and PR=2,662 feet, nearly. 36. With a view to determine the distance and height of a rock seen rising from the sea, remote from the shore, two stations are taken along the shore, both on the high water mark. The direct distance between the two stations, being measured with a chain, is 2,527 links; and the two horizontal angles taken at the two stations (as described in Ex. 34), are 89° 15' and 86° 21'. The angle of elevation of the highest point of the rock, taken at the first station, being 1° 48'. What is the distance of the top of the rock from the first station, in miles, and its height, in feet, above the high water level, taking 5 feet as the height of the instrument, and regarding the earth as a plane ? Ans. Distance, 4:11 + miles; height, 687 — feet. 41. A meteorologist was desirous of ascertaining, as nearly as he could, the general height of the clouds on a particular occasion. As the wind blew gently and steadily, he first obtained their velocity from that of their shadows, and found that they travelled at about the rate of 15 miles per hour. He then marked the exact time at which the margin of a cloud was right over head; and precisely 8 minutes after, observed that its angle of elevation was about 31°. What conclusion could he draw from that observation, no allowance being made for the Earth's curvature, which would make no difference worth estimating? Ans. The height of the cloud was about 1} mile. 43. On another occasion the height of the clouds was tried by a different method, their velocity not being regarded as uniform. A station was taken on an eminence overlooking an extensive plain. The observer waited till he perceived a small cloud, or rather part of a cloud, vertically over the station. He watched the same cloud, keeping the telescope of the theodolite directed to it, tili its shadow touched a suitable point on the plain, a friend keeping his eye on the shadow while he observed the cloud. He then took its angular elevation, which was 65o. He still kept the telescope upon it till its shadow touched
represent PR), with the horizontal lines drawn on the surface of the board, and with threads for the oblique lines; or simply a piece of wood cut into the form EPRF. (The latter will be found in the set of Models prepared for the illustration of this work, and sold by the publishers, Messrs Sutherland and Knox.) Page 10
another convenient point on the plain, and then found its elevation 42°. He next measured the distance between the two points on the plain, which proved to be 923 yards. What was the height of the cloud above the station ? Ans. 1,433 yards, or •814 of a mile. 44. Taking the Earth as a sphere of 7,912 miles in diameter, what will be the dip of the sea-horizon as seen om a mountain 3 miles in height, making no allowance for terrestrial refraction ? Ans. 2° 14'.
NOTE. By the dip of the sea-horizon is meant the angle of depression at which the extreme surface of the sea appears below a horizontal line or plane at the point of observation.
51. The distances of three stations, P, Q, R, from each other, are as follows: PQ=4, QR= 5, PR=6, chains. A point E is in the same line with PR and at such a distance from P that the angle QER is of 36o. What is the distance of E from P, Q, and R? Ans. 2:30+, 5.63 –, and 8:30+, chains.
EXERCISES IN CHAPTER II, PROBLEM VI.
1. What are the logarithms of the numbers 26,583 ; 8,425.9; 164:51; and 98.506 ? 2. Of 4.387,28; 6-3; 305.472; and 9,845.54 ?
EXERCISES IN CHAPTER II, PROBLEM IX.
1. Find the natural numbers corresponding to the logarithms 2.000000; 3.220108; 5:831166; and 1.597146. 2. Of 4.820136 ; and 2:271377. 3. Of 1.371070; 2:472440 ; 0.837958; and 3.450100. EXERCISES IN CHAPTER II, PROBLEM X.
Find the following products by logarithms :1. 7 x 6 x 5. 2. 258.7 x 6:844 3. 5.48 x 6.72 x 3.81 x 2.99. 4. 825 × 351 x 26•2 x 3.08, completing with ciphers. 5. 208.91 x 34.228. (See Example 2.) 6. 5.3 x 2.8. 7. 48.574 x 5.8347 x 12.1499. 8. 786.6 x .0428. (See Note 1.) 9. •004,99 x .658. 10. 67.433 x 5.8628 x 38 x.006.
EXERCISES IN CHAPTER II, PROBLEM XI.
1. Divide 3,312 by 92. 2. Divide 288 by 6.4. 3. Divide 92:61 by 44:1. 4. Divide 909.23 by 7.4888. 5. Divide 16:56 by 46. (See Note 1.) 6. Divide 6.8544 by 329.89. 7. Divide 4.83 by 0.882. (See Note 2.) 8. Divide 0.6076 by .0084. 9. Divide 0.882 by 4.833. (See Note 3.) 10. Divide .0084 by 0:6076.
EXERCISES IN CHAPTER II, PROBLEM XII. Compute the fourth term of each of the following proportions, by logarithms : 1. As 279 : 651 :: 657 : 2. As 94.25 : 486:3 :: 8.828 : 3. As 3.2496 : 54.853 :: 15 : 4. As 0.28 : 6.75 :: 4.838 : (See Note 2.) 5. As ·0656 : 0835 :: 2:24 : 6. As .00958 : :06534 :: .02755 : 7. As 25 : 0.669 :: 48 : (See Note 3.) 8. As 94:3 : 94.2 :: 08886 : 9. As 0:628 : 0.723 :: .05 : 10. As 2 : 0.66 :: .048 : (See Note 4.) 11. As 0.484,66 : .058,32 :: :079,63 : 12. As 892 : 24:4 :: 8:52 : (See Note 5.) 13. As 73.598 : 0:694 :: 058 :
EXERCISES IN CHAPTER II, PROBLEM XIII.
1. Square the numbers 25; 38.3; 5.4962; and 48:38. 2. Cube the numbers 15; 6:3; and 18.499. 3. Square 384.5, and cube 198, completing the results with ciphers. 4. Square 0.852,63 ; 0179; and .008. (See the Note.) 5. Cube 0.2835 and .0899.
EXERCISES IN CHAPTER II, PROBLEM XV.
1. Extract the square roots of 84,100; 88:36; 285,168 ; and 10. 2. Extract the cube roots of 6,859 ; 3,375,000; 857.4 ; and 600. 3. Find the square roots of .0081 ; 0.658; .00055 ; and 0.1. (See Note.) 4. Calculate the cube roots of .000,585 ; 0:5 ; .0688 ; and 0.8.
EXERCISES IN CHAPTER III.
1. An ancient Roman mile was equivalent to 1612 yards of British Imperial Measure. Convert 556 Roman miles into Imperial miles. 2. If 8.546 acres of land are valued at £564, 10s., what should be the value of 2.868 acres, at the same rate ? 3. What is the interest of £457, 8s., for 7 months, at 44 per cent. per annum ? 4. Compute the interest of £64:10: 9, for 288 days at 33 per cent. per annum. 5. Calculate the amount of £10, for fifty years, at 4 per cent. per annum, compound interest. 6. What sum will one penny amount to in 500 years, if accumulating, all the while, at 5 per cent. per annum, compound interest? 7. What is the compound interest of £383 for 5) years at 31 per cent. per annum ? 8. Find the diameter of a circle, the circumference of which is 408.2. 9. What is the area, in acres, of a circle whose radius is 2.85 chains ?
10. Determine the area of an ellipse, the transverse axis being 7.38, and the conjugate 4:71. 11. Compute the number of cubic feet in a cone, whose base is 356 inches in diameter, and whose height is 98 inches. 12. What is the weight of an ivory ball 14 inches in diameter, at the rate of 1820 ounces (Av.) to the cubic foot ?
EXERCISES IN CHAPTER VI, PROBLEM I.
1. What are the log. sines of 24°, of 68°, of 35° 32', and of 84° 15' ? 2. What is the log. tangent of 32° 12', and that of 48° 40'? 3. What is the log. cosine of 18° 24', and the log. secant of 50° 8' ?
EXERCISES IN CHAPTER VI, PROBLEM II. Find the log. sine of 22° 48' 54', the log. tangent of 63° 25' 16", the log. secant of 38° 14' 5", and the log. cosine of 76° 0' 48".
EXERCISES IN CHAPTER VI, PROBLEM V.
What are the log. sines of 104° 18', 126° 12' 24", and 164° 54' 48" ?
EXERCISES IN CHAPTER VI, PROBLEM VII.
What angles correspond to log. sine 9.520271, log. tangent 10•407574, and log. secant 10:566325 ? 2. Find the numbers of degrees and minutes corresponding to log. sine 9.284776, log. tan 10550548, and log. secant 11:132496. 3. What are the angles answering to log. cosine 9.995555, to log. cotan 10:508800, and to log. cosecant 10.030568 ?
EXERCISES IN CHAPTER VII, PROBLEM I. 1. The acute angle at the base of a right-angled triangle is 182° : what is that at the vertex ?
2. The acute angle at the vertex being 84° 133', what is that at the base ? 3. If the one acute angle contain 28° 44' 55", how many degrees are in the other ?
EXERCISES IN CHAPTER VII, PROBLEM II.
1. Hyp. = 5438. Acute angle at base = 35°. Perp. required. 2. Hyp. = 2:49. Acute angle at vertex = 43° 25'. Base required. 3. Hyp. = 1000. One acute angle = 36° 16'. What are the legs ? 4. Hyp. = .04. Angle at base 43° 40'. Sides required. 5. Hyp. = 8.2958. Acute angle at vertex = 54° 23'. Perp. required.
EXERCISES IN CHAPTER VII, PROBLEM III.
1. Base = 845. Adjacent acute angle = 28° 25'. Hyp. required. 2. Perp. = 18:48. Vertical angle = 59° 12. Hyp. required. 3. Base = 4.843. Vertical angle = 7o. Hyp. required. 4. Compute, by Trigonometry, the diagonal of a square whose side is 749. 5. Perp. = 583.26. Acute angle at base 18° 481 Hyp. required.
EXERCISES IN CHAPTER VII, PROBLEM IV.
1. Base = 14:04. Acute angle adjacent = 72° 48'. Perp. required. 2. Perp. = 3 ft. 4 in. Vertical angle=22° 28'. Base required. 3. Base = 384. Vertical angle = 48° 26' 12". Perp. required. 4. Base = 0·440,83. Acute angle at the base : 33° 38.2'. Perp. required. Page 11
7. E by N, 86 miles ......... ? 8. ..... between S and W, 200 miles........844 miles : , ? 10. ...315............10° 38': ? 11. 87 feet, ..... .19° 59'. ? 12. 12 feet...... 25 feet......30 inches lower......? 13. .....Latitude 51° 25'............ .75 feet ...........? 16. ....... 24 feet square.......36 yards...... 33° 39'...... 161 feet.........? 17. .28° 38'.......5 feet ... .......12° 12' .......496 feet. .? 18. ....101 feet...... 34° 45'......20 feet above the base .? 19. ..... .8° 14' ;......... 78° 22' ..........168 feet, ......? 20. ......N 22° E, ......N 28 W......300......200...... ? 21. ........ ESE ;........E by N........SW by S........500 yards ..........? 24. .........108.........angular depression of the top...... 2° 49', and of the base 26° 52' ? 26. .........1 mile..........: ZABC=54° 16'. ZDBC= 64° 20'. <BAC = 73° 28'. ZODE= 58° 59'. BCD= 61° 33'. ZDCE=56° 11'. 27. ... 15° 31'.........300 feet......... 21° 54'..... 29. ......6° 3' and 8° 30'......8 chains......0° 44'.... the more distant station being the lower. 30. .......8 chains.......3° 50' and 7° 29'....... 26 feet higher......12 feet higher......? 31. ......5 feet......1 chain......38° 10'......1 chain...... ? 32. ......... The base EF of 100 feet.........86° 18' and 70° 47' ; and the angle PER, 15° 42' ......? 34. ....... a base of 30 chains....... angle of elevation at E = 2° 53'. Horizontal angle at E= 86o. Horizontal angle at F= 91° 13'......... 36. ........ .12.88 chains ;. .91° 36' and 84° 2'. .2° 12'. 41. ...........36 miles per hour ...........5 minutes after .......... 24° 42'. What was the height of the cloud in feet? 43. ......... 71°.........45°.........657 yards. What was the height of the cloud in feet? 44. ..2 miles............? 51. .PQ = 2, QR = 3, PR = 4.........45° ........ ? Page 12
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180 5273 5514 5755 5996 6237 | 6477 6718 6958 7198 7439 181 7679 7918 8158 8398 8637 8877 9116 9355 9594 9833 182 260071 0310 0548 0787 1025 1263 1501 | 1739 1976 2214 183 2451 2688 2925 3162 3399 3636 3873 4109 4346 4582 184 4818 5054 5290 5525 5761 5996 6232 6467 | 6702 6937 185 7172 7406 7641 7875 8110 8344 8578 8812 9046 9279 186 9513 9746 9980 0213 0446 0679 0912 1144 1377 1609 187 271842 2074 2306 2538 2770 3001 3233 3464 3696 3927 188 4158 4389 4620 4850 5081 5311 5542 5772 60026232 189 6462 6692 6921 7151 7380 7609 7838 80678296 8525 190 8754 8982 9211 9439 9667 9895 0123 0351 0578 0806 191 2810331261 1488 1715 1942 2169 2396 2622 2849 3075 192 3301 3527 3753 3979 4205 4431 4656 4882 5107 5332 193 5557 5782 6007 6232 6456 6681 6905 7130 7354 7578 194 7802 8026 8249 84738696 | 89209143 9366 9589 9812 195 | 290035 0257 0480 0702 0925 1147 | 1369 1591 | 1813 2034 196 2256 2478 2699 2920 3141 3363 3584 3804 4025 4246 197 4466 4687 4907 5127 5347 5567 5787 6007 6226 | 6446 198 6665 6884 7104 7323 7542 7761 7979 8198 8416 8635 199 8853 9071 | 9289 9507 9725 9943 0161 0378 0595 0813
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965 4527 4572 | 4617 4662 4707 || 4752 4797 4842 4887 4932 966 4977 5022 5067 51125157 5202 5247 5292 5337 5382 967 5426 5471 5516 5561 5606 5651 5696 5741 5786 5830 968 5875 5920 5965 6010 6055 6100 6144 6189 6234 6279 969 6324 6369 6413 6458 6503 6548 6593 663766826727 970 6772681768616906 6951 6996 7040 7085 71307175 971 7219 7264 73097353 7398 7443 7488 7532 7577 7622 972 7666 7711 7756 7800 7845 | 7890 7934 7979 8024 8068 973 8113 8157 8202 | 8247 8291 8336 8381 8425 8470 8514 974 8559 8604 8648 8693 8737 | 87828826 8871 8916 8960 975 9005 9049 9094 9138 9183 | 9227 9272 9316 | 9361 9405 976 9450 9494 9539 9583 9628 | 9672 97:7 9761 9806 9850 977 9895 9939 9983 0028 0072 || 011701610206 0250 0294 978 990339 0383 0428 0472 0516 || 0561 0605 | 0650 0694 0738 979 0783 0827 0871 0916 0960 1004 1049 | 1093 | 1137 | 1182 980 1226 1270 1315 1359 1403 1448 | 1492 1536 1580 1625 981 1669 | 1713 1758 1802 1846 1890 | 1935 | 1979 2023 2067 982 2111 2156 2200 2244 2288 2333 2377 2421 2465 2509 983 2554 2598 2642 26862730 || 2774 2819 2863 2907 2951 984 2995 3039 3083 3127 3172 || 3216 3260 3304 3348 3392 985 3436 3480 3524 3568 3613 3657 3701 3745 3789 3833 986 3877 3921 3965 4009 4053 4097 4141 | 4185 4229 4273 987 4317 4361 4405 4449 4493 4537 4581 4625 4669 4713 988 4757 4801 4845 4889 4933 | 4977 5021 5065 5108 5152 989 5196 5240 5284 5328 5372 || 5416 5460 5504 5547 5591 990 5635 5679 | 5723 5767 5811 5854 5898 5942 59866030 991 6074 6117 6161 | 6205 6249 6293 6337 638064246468 992 65126555 6599 6643 6687 67316774 6818 6862 | 6906 993 6949 6993 7037 7080 7124 | 7168 7212 7255 7299 7343 994 7386 7430 7474 7517 7561 7605764876927736 7779 995 7823 7867 7910 7954 7998 8041 8085 | 8129 81728216 996 8259 8303 83478390 8434 8477 8521 8564 8608 8652 997 8695 873987828826 8869 8913 8956 9000 9043 9087 998 9131 9174 9218 9261 9305 9348 9392 9435 94799522 999 9565 96099652 9696 9739 9783 9826 9870 9913 9957 Page 16
6° Sine. Cosec. Tang. Cotang. Secant. Cosine. 09.019235 10.980765 9.021620 10.978380 10.002386 9.997614 60 020435 979565 022834 977166 002399 997601 59 021632 978368 024044 975956 002412 997588 58 022825 977175 025251 974749 002426 997574 57 024016 975984 026455 973545 002439 997561 56! 9.025203 10.974797 9.027655 10.972345 10.002453 9.997547 155 026386 973614 028852 971148 002466 997534 54 027567 972433 030046 969954 002480 997520 53 028744 971256 031237 968763 002493 997507 52 9 029918 970082 032425 967575 002507 997493 51 109.031089 10.968911 9.033609 10.966391 10.002520 9.997480 50 11 032257 967743 034791 965209 002534 997466 49 12 033421 966579 035969 964031 002548 997452 48 13 034582 965418 037144 962856 002561 997439 47 14 035741 964259 038316 961684 002575 997425 46 15 9.036896 10.963104 9.039485 10.960515 10.002589 9.997411 45 16 038048 961952 040651 959349 002603 997397 44 17 039197 960803 041813 958187 002617 997383 43 18 040342 959658 042973 957027 002631 997369 42 19 041485 958515 044130 955870 002645 997355 41 20 9.042625 10.957375 9.045284 10.954716 10.002659 9.997341 40 21 043762 956238 046434 953566 002673 997327 39 22 044895 955105 047582 952418 002687 997313 38 046026 953974 048727 951273 002701 997299 37 124 047154 952846 049869 950131 002715 997285 36 25 9.048279 10.951721 9.051008 10.948992 10.002729 9.997271 35 049400 950600 052144 947856 002743 997257 34 27 050519 949481 053277 946723 002758 997242 33 28 051635 948365 054407 945593 002772 997228 32 29 052749 947251 055535 944465 002786 997214 31 30 9.053859 10.946141 9.056659 10.943341 10.002801 9.997199 30 31 054966 945034 057781 942219 002815 997185 29 32 056071 943929 058900 941100 002830 997170 281 33 057172 942828 060016 939984 002844 997156 27 34 058271 941729 061130 938870 002859 997141 26 35 9.059367 10.940633 9 062240 10.937760 10.002873 9.997127 25 136 060460 939540 063348 936652 002888 997112 24 137 061551 938449 064453 935547 002902 997098 23 062639 937361 065556 934444 002917 997083 22 063724 936276 066655 933345 002932 997068 21 40 9.064806 10.935194 9.067752 10.932248 10.002947 9.997053 20 41 065885 934115 068846 931154 002961 997039 19 42 066962 933038 069938 930062 002976 997024 18 43 068036 931964 071027 928973 002991 997009 17 44 069107 930893 072113 927887 003006 996994 16 45 9.070176 10.929824 9.073197 10.926803 10.003021 9.996979 15 146 071242 928778 074278 925722 003036 996964 14 47 072306 927694 075356 924644 003051 996949 13 48 073366 926634 076432 923568 003066 996934 12 49 074424 925576 077505 922495 003081 996919 11 150 9.075480 10.924520 9.078576 10.921424 10.003096 9.996904 10 51 076533 923467 079644 920356 003111 9968899 152 077583 922417 080710 919290 003126 996874 53. 078631 921369 081773 918227 003142 996858 7 54 079676 920324 082833 917167 003157 9968436 55 9.080719 10.919281 9.083891 10.916109 | 10.003172 9.996828 56 081759 918241 084947 915053 003188 996812 4 57 082797 917203 086000 914000 003203 996797 58 083832 916168 087050 912950 003218 9967822 159 084864 915136 088098 911902 003234 9967661 60 085894 914106 089144 910856 003249 996751 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 17
8° Sine. Cosec. Tang. Cotang. Secant. Cosine. 09.143555 10.856445 9.147803 10.852197 10.004247 9.995753 60 144453 855547 148718 851282 004265 995735 59 145349 854651 149632 850368 004283 995717 58 3 146243 853757 150544 849456 004301 995699 57 4 147136 852864 151454 848546 004319 995681 56 9.148026 10.851974 9.152363 10.847637 10.004336 9.995664 55 6 148915 851085 153269 846731 004354 995646 54 149802 850198 154174 845826 004372 995628 53 8 150686 849314 155077 844923 004390 995610 52 151569 848431 155978 844022 004409 995591 51 10 9.152451 10.847549 9.156877 10.843123 10.004427 9.995573 50 11 153330 846670 157775 842225 004445 995555 49 12 154208 845792 158671 841329 004463 995537 48 13 155083 844917 159565 840435 004481 995519 47 14 155957 844043 160457 839543 004499 995501 46 15 9.156830 10.843170 9.161347 10.838653 10.004518 9.995482 45 16 157700 842300 162236 837764 004536 995464 44 17 158569 841431 163123 836877 004554 995446 43 18 159435 840565 164008 835992 004573 995427 42 19 160301 839699 164892 835108 004591 995409 41 20 9.161164 10.838836 9.165774 10.834226 10.004610 9.995390 40 21 162025 837975 166654 833346 004628 995372 39 22 162885 837115 167532 832468 004647 995353 38 23 163743 836257 168409 831591 004666 995334 37 24 164600 835400 169284 830716 004684 995316 36 25 9.165454 10.834546 9.170157 10.829843 10.004703 9.995297 35 26 166307 833693 171029 828971 004722 995278 34 27 167159 832841 171899 828101 004740 995260 33 128 168008 831992 172767 827233 004759 995241 32 29 168856 831144 173634 826366 004778 995222 31 30 9.169702 10.830298 9:174499 10.825501 10.004797 9.995203 30 31 170547 829453 175362 824638 004816 995184 29 32 171389 828611 176224 823776 004835 995165 28 33 172230 827770 177084 822916 004854 995146 27 34 173070 826930 177942 822058 004873 995127 26 35 9.173908. 10.826092 9.178799 10.821201 10.004892 9.995108 25 36 174744 825256 179655 820345 004911 995089 24 37 175578 824422 180508 819492 004930 995070 23 38 176411 823589 181360 818640 004949 995051 22 39 177242 822758 182211 817789 004968 995032 21 40 9.178072 10.821928 9.183059 10.816941 10.004987 9.995013 20 41 178900 821100 183907 816093 005007 994993 19 42 179726 820274 184752 815248 005026 994974 18 43 180551 819449 185597 814403 005045 994955 17 144 181374 818626 186439 813561 005065 994935 16 45 9.182196 10.817804 9.187280 10.812720 10.005084 9.994916 15 46 183016 816984 188120 811880 005104 994896 14 47 183834 816166 188958 811042 005123 994877 13 48 184651 815349 189794 810206 005143 994857 12 49 185466 814534 190629 809371 005162 994838 11 50 9.186230 10.813720 9.191462 10.808538 10.005182 9.994818 10 51 187092 812908 192294 807706 005202 994798 9 52 187903 812097 193124 806876 005221 994779 8 53 188712 811288 193953 806047 005241 994759 7 54 189519 810481 194780 805220 005261 994739 55 9.190325 10.809675 9.195606 10.804394 10.005281 9.994719 56 191130 808870 196430 803570 005300 994700 57 191933 808067 197253 802747 005320 994680 58 192734 807266 198074 801926 005340 994660 59 193534 806466 198894 801106 005360 994640 60 194332 805668 199713 800287 005380 994620 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 18
12° Sine. Cosec. Tang. Cotang. Secant. Cosine. 09.317879 10.682121 9.327475 10.672525 10.009596 9.990404 60 1 318473 681527 328095 671905 009622 990378 59 2 319066 680934 328715 671285 009649 990351 58 3 319658 680342 329334 670666 009676 990324 57 320249 679751 329953 670047 009703 990297 56 59.320840 10.679160 9.330570 10.669430 10.009730 9.990270 155 6 321430 678570 331187 668813 009757 990243 54 322019 677981 331803 668197 009785 990215 53 8 322607 677393 332418 667582 009812 990188 52 323194 676806 333033 666967 009839 990161 51 10 9.323780 10.676220 9.333646 10.666354 10.009866 9.990134 50 11 324366 675634 334259 665741 009893 990107 49 12 324950 675050 334871 665129 009921 990079 48 13 325534 674466 335482 664518 009948 990052 47 14 326117 673883 336093 663907 009975 990025 46 15 9.326700 10.673300 9.336702 10.663298 10.010003 9.989997 45 16 327281 672719 337311 662689 010030 989970 44 17 327862 672138 337919 662081 010058 989942 43 18 329442 671558 338527 661473 010085 989915 42 19 329021 670979 339133 660867 010113 989887 41 20 9.329599 10.670401 9.339739 10.660261 10.010140 9.989860 40 21 330176 669824 340344 659656 010168 989832 39 22 330753 669247 340948 659052 010196 989804 38 23 331329 668671 341552 658448 010223 989777 37 24 331903 668097 342155 657845 010251 989749 36 25 9.332478 10.667522 9.342757 10.657243 10.010279 9.989721 35 26 333051 666949 343358 656642 010307 989693 34 27 333624 666376 343958 656042 010335 989665 33 28 334195 665805 344558 655442 010363 989637 32 29 334766 665234 345157 654843 010391 989609 31 30 9.335337 10.664663 9.345755 10.654245 10.010418 9.989582 30 31 335906 664094 346353 653647 010447 989553 29 32 336475 663525 346949 653051 010475 989525 28 33 337043 662957 347545 652455 010503 989497 27 34 337610 662390 348141 651859 010531 989469 26 35 9.338176 10.661824 9.348735 10.651265 10.010559 9.989441 25 36 338742 661258 349329 650671 010587 989413 24 37 339307 660693 349922 650078 010615 989385 23 339871 660129 350514 649486 010644 989356 22 39 340434 659566 351106 648894 010672 989328 21 40 9.340996 10.659004 9.351697 10.648303 10.010700 9.989300 20 341558 658442 352287 647713 010729 989271 19 42 342119 657881 352876 647124 010757 989243 18 43 342679 657321 353465 646535 010786 989214 17 44 343239 656761 354053 645947 010814 989186 16 45 9.343797 10.656203 9.354640 10.645360 10.010843 9.989157 15 46 344355 655645 350227 644773 010872 989128 14 47 344912 655088 355813 644187 010900 989100 13 48 345469 654531 356398 643602 010929 989071 12 49 346024 653976 356982 643018 010958 989042 11 50 9.346579 10.653421 9.357566 10.642434 10.010986 9.989014 10 51 347134 652866 358149 641851 011015 988985 52 347687 652313 358731 641269 011044 988956 53 348240 651760 359313 640687 011073 988927 54 348792 651208 359893 640107 011102 988898 55 9.349343 10.650657 9.360474 10.639526 10.011131 9.988869 56 349893 650107 361053 638947 011160 988840 57 350443 649557 361632 638368 011189 988811 58 350992 649008 362210 637790 011218 988782 591 351540 648460 362787 637213 011247 988753 60 352088 647912 363364 636636 011276 988724 0 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 19
20° Sine. Cosec. Tang. Cotang. Secant. Cosine. 09.534052 10.465948 9.561066 10.438934 10.027014 9.972986 60 1 534399 465601 561459 438541 027060 972940 59 2 534745 465255 561851 438149 027106 972894 58 3 535092 464908 562244 437756 027152 972848 57 4 535438 464562 562636 437364 027198 972802 56 5 9.535783 10.464217 9.563028 10.436972 10.027245 9.972755 55 6 536129 463871 563419 436581 027291 972709 54 7 536474 463526 563811 436189 027337 972663 53 8 536818 463182 564202 435798 027383 972617 52 9 537163 462837 564592 435408 027430 972570 51 10 9.537507 10.462493 9.564983 10.435017 10.027476 9.972524 50 11 537851 462149 565373 431627 027522 972478 49 12 538194 461806 565763 434237 027569 972431 48 13 538538 461462 566153 433847 027615 972385 47 14 538880 461120 566542 433458 027662 972338 46 15 9.539223 10.460777 9.566932 10.433068 10.027709 9.972291 45 16 539565 460435 567320 432680 027755 972245 44 17 539907 460093 567709 432291 027802 972198 43 18 540249 459751 568098 431902 027849 972151 42 19 540590 459410 568486 431514 027895 972105 41 20 9.540931 10.459069 9.568873 10.431127 10.027942 9.972058 40 21 541272 458728 569261 430739 027989 972011 39 22 541613 458387 569648 430352 028036 971964 38 23 541953 458047 570035 429965 028083 971917 37 24 542293 457707 570422 429578 028130 971870 36 25 9.542632 10.457368 9.570809 10.429191 10.028177 9.971823 35 26 542971 457029 571195 428805 028224 971776 34 27 543310 456690 571581 428419 028271 971729 33 28 543649 456351 571967 428033 028318 971682 32 29 543987 456013 572352 427648 028365 971635 31 30 9.544325 10.455675 9.572738 10.427262 10.028412 9.971588 30 31 544663 455337 573123 426877 028460 971540 29 32 545000 455000 573507 426493 028507 971493 28 33 545338 454662 573892 426108 028554 971446 27 34 545674 454326 574276 425724 028602 971398 26 35 9.546011 10.453989 9.574660 10.425340 10.028649 9.971351 25 36 546347 453653 575044 424956 028697 971303 24 37 546683 453317 575427 424573 028744 971256 23 38 547019 452981 575810 424190 028792 971208 22 39 547354 452646 576193 423807 028839 971161 21 40 9.547689 10.452311 9.576576 10.423424 10.028887 9.971113 20 41 548024 451976 576959 423041 028934 971066 19 42 548359 451641 577341 422659 028982 971018 18 43 548693 451307 577723 422277 029030 970970 17 44 549027 450973 578104 421896 029078 970922 16 45 9.549360 10.450640 9.578486 10.421514 10.029126 9.970874 15 46 549693 450307 578867 421133 029173 970827 14 47 550026 449974 579248 420752 029221 970779 131 48 550359 449641 579629 420371 029269 970731 12 49 550692 449308 580009 419991 029317 970683 11 150 9.551024 10.448976 9.580389 10.419611 10.029365 9.970635 10 51 551356 448644 580769 419231 029414 9705869 52 551687 448313 581149 418851 029462 970538 8 53 552018 447982 581528 418472 029510 970490 54 552349 447651 581907 418093 029558 970442 55 9.552680 10.447320 9.582286 10.417714 10.029606 9.970394 56 553010 446990 582665 417335 029655 970345 57 553341 446659 583043 416957 029703 970297 58 553670 446330 593422 416578 029751 979249 59 554000 446000 583800 416200 029800 970200 1 160 554329 445671 584177 415823 029848 970152 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 20
23° Sine. Cosec. Tang. Cotang: Secant Cosine. 09.59 1878 10.408122 9.627852 10.372148 10.035974 9.964026 60 1 592176 407824 628203 371797 036028 963972 59 592473 407527 628554 371446 036081 963919 58 592770 407230 628905 371095 036135 963865 57 593067 406933 629255 370745 036189 963811 56 9.593363 10.406637 9.629606 10.370394 10.036243 9.963757 55 593659 406341 629956 370044 036296 963704 54 7 593955 406045 630306 369694 036350 963650 53 594251 405749 630656 369344 036404 963596 52 9 594547 405453 631005 368995 036458 963542 51 110 9.594842 10.405158 9.631355 10.368645 10.036512 9.963488 150 11 595137 404863 631704 368296 036566 963434 491 12 595432 404568 632053 367947 036621 963379 48 13 595727 404273 632401 367599 036675 963325 47 14 596021 403979 632750 367250 036729 963271 46 15 9.596315 10.403685 9.633098 10.366902 (10.036783 9.963217 45 596609 403391 633447 366553 036837 963163 44 17 596903 403097 633795 366205 036892 963108 43 18 597196 402804 634143 365857 036946 963054 42 19 597490 402510 634490 365510 037001 962999 41 20 9.597783 10.402217 9.634838 10.365162 10.037055 9.962945 40 21 598075 401925 635185 364815 037110 962890 39 122 598368 401632 635532 364468 037164 962836 38 23 598660 401340 635879 364121 037219 962781 37 124 598952 401048 636226 363774 037273 962727 36 25 9.599244 10.400756 9.636572 10.363428 10.037328 9.962672 35 26 599536 400464 636919 363081 037383 962617 34 27 599827 400173 637265 362735 037438 962562 33 28 600118 399882 637611 362389 037492 962508 32 29 600409 399591 637956 362044 037547 962453 31 30 9.600700 10.399300 9.638302 10.361698 10.037602 9.962398 30 31 600990 399010 638647 361353 037657 962343 29 32 601280 398720 638992 361008 037712 962288 28 33 601570 398430 639337 360663 037767 962233 27 134 601860 398140 639682 360318 037822 962178 26 35 9.602150 10.397850 9.640027 10.359973 10.037877 9.962123 25 36 602439 397561 640371 359629 037933 962067 24 602728 397272 640716 359284 037988 962012 23 603017 396983 641060 358940 038043 961957 22 39 603305 396695 641404 358596 038098 961902 21 40 9.603594 10.396406 9.641747 10.358253 10.038154 9.961846 20 41 603882 396118 642091 357909 038209 961791 19 42 604170 395830 642434 357566 038265 961735 18 43 604457 395543 642777 357223 038320 961680 17 44 604745 395255 643120 356880 038376 961624 16 45 9.605032 10.394968 9.643463 10.356537 10.038431 9.961569 15 46 605319 394681 643806 356194 038487 961513 14 47 605606 394394 644148 355852 038542 961458 13 48 605892 394108 644490 355510 038598 961402 12 49 606179 393821 644832 355168 038654 961346 11 50 9.606465 10.393535 9.645174 10.354826 10.038710 9.961290 10 51 606751 393249 645516 354484 038765 9612359 52 607036 392964 645857 354143 038821 9611798 153 607322 392678 646199 353801 038877 961123 154 607607 392393 646540 353460 038933 961067 6 55 9.607892 10.392108 9.646881 10.353119 10.038989 9.961011 5 156 608177 391823 647222 352778 039045 960955 4 157 608461 391539 647562 352438 039101 960899 3 58 608745 391255 647903 352097 039157 960843 59 609029 390971 648243 351757 039214 960786 60 609313 390687 648583 351417 039270 960730 Cosine. Secant Cotang. Tang. Cosec. Sine. Page 21
28° Sine. Cosec. Tang. Cotang: Secant. Cosine. 0 9.671609 10.328391 9.725674 10.274326 10.054065 9.945935 60 1 671847 328153 725979 274021 054132 945868 59 2 672084 327916 726284 273716 054200 945800 58 3 672321 327679 726588 273412 054267 945733 57 4 672558 327442 726892 273108 054334 945666 56 5 9.672795 10.327205 9.727197 10.272803 10.054402 9.945598 55 6 673032 326968 727501 272499 054469 945531 154 71 673268 326732 727805 272195 054536 945464 53 8 673505 326495 728109 271891 054604 945396 52 9 673741 326259 728412 271588 054672 945328 51 109.673977 10.326023 9.728716 10.271284 10.054739 9.945261 50 11 674213 325787 729020 270980 054807 945193 49 12 674448 325552 729323 270677 054875 945125 48 13 674684 325316 729626 270374 054942 945058 47 14 674919 325081 729929 270071 055010 944990 46 15 9.675155 10.324845 9.730233 10.269767 10.055078 9.944922 45 16 675390 324610 730535 269465 055146 944854 44 17 675624 324376 730838 269162 055214 944786 43 18 675859 324141 731141 268959 055282 944718 42 19 676094 323906 731444 268556 055350 944650 41 20 9.676328 10.323672 9.731746 10.268254 | 10.053418 9.944582 40 21 676562 323438 732048 267952 055486 944514 139 22 676796 323204 732351 267649 055554 944446 38 23 677030 322970 732653 267347 055623 944377 37 24 677264 322736 732955 267045 055691 944309 36 25 9.677498 10.322502 9.733257 10.266743 10.055759 9.944241 35 26 677731 322269 733558 266442 055828 944172 34 27 677964 322036 733860 266140 055896 944104 33 28 678197 321803 734162 265838 055964 944036 32 29 678430 321570 734463 265537 056033 943967 31 309.678663 10.321337 9.734764 10.265236 10.056101 9.943899 30 31 678895 321105 735066 264934 056170 943830 29 32 679128 320872 735367 264633 056239 943761 28 33 679360 320640 735668 264332 056307 943693 27 34 679592 320408 735969 264031 056376 943624 26 35 9.679824 10.320176 9.736269 10.263731 10.056445 9.943555 25 136 680056 319944 736570 263430 056514 943486 24 37 680288 319712 736871 263129 056583 943417 23 38 680519 319481 737171 262829 056652 943348 22 139 680750 319250 737471 262529 056721 943279 21 409.680982 10.319018 9.737771 10.262229 10.056790 9.943210 20 41 681213 318787 738071 261929 056859 943141 19 42 681443 318557 738371 261629 056928 943072 18 43 681674 318326 738671 261329 056997 943003 17 44 681905 318095 738971 261029 057066 942934 16 45 9.682135 10.317865 9.739271 10.260729 10.057136 9.942864 15 46 682365 317635 739570 260430 057205 942795 14 47 682595 317405 739870 260130 057274 942726 13 48 682825 317175 740169 259831 057344 942656 12 49 683055 316945 740468 259532 057413 942587 11 50 9.683284 10.316716 9.740767 10.259233 10.057483 9.942517 10 51 683514 316486 741066 258934 057552 942448 9 52 683743 316257 741365 258635 057622 942378 8 153 683972 316028 741664 258336 057692 942308 54 684201 315799 741962 258038 057761 942239 55 9.684430 10.315570 9.742261 10.257739 10.057831 9.942169 56 684658 315342 742559 257441 057901 942099 57 684887 315113 742858 257142 057971 942029 58 685115 314885 743156 256844 058041 941959 2 59 685343 314657 743454 256546 058111 941889 685571 314429 743752 256248 058181 941819 0 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 22
36° Sine. Cosec. Tang. Cotang. Secant. Cosine. 09.769219 10.230781 9.861261 10.138739 10.092042 9.907958 60 769393 230607 861527 138473 092134 907866 59 2 769566 230434 861792 138208 092226 907774 58 3 769740 230260 862058 137942 092318 907682 57 4 769913 230087 862323 137677 092410 907590 56 5 9.770087 10.229913 9.862589 10.137411 10.092502 9.907498 55 6 770260 229740 862854 137146 092594 907406 54 7 770433 229567 863119 136881 092686 907314 53 8 770606 229394 863385 136615 092778 907222 52 9 770779 229221 863650 136350 092871 907129 51 10 9.770952 10.229048 9.863915 10.136085 10.092963 9.907037 50 11 771125 228875 864180 135820 093055 906945 49 12 771298 228702 864445 135555 093148 906852 48 13 771470 228530 864710 135290 093240 906760 47 771643 228357 864975 135025 093333 906667 46 15 9.771815 10.228185 9.865240 10.134760 10.093425 9.906575 45 116) 771987 228013 865505 134495 093518 906482 44 17 772159 227841 865770 134230 093611 906389 43 18 772331 227669 866035 133965 093704 906296 42 19 772503 227497 866300 133700 093796 906204 41 209.772675 10.227325 9.866564 10.133436 10.093889 9.906111 40 21 772847 227153 866829 133171 093982 906018 39 22 773018 226982 8670.4 132906 094075 905925 38 773190 226810 867358 132642 094168 905832 37 24 773361 226639 867623 132377 094261 905739 36 25 9.773533 10.226467 9.867887 10.132113 10.094355 9.905645 35 26 773704 226296 868152 131848 094448 905552 34 27 773875 226125 868416 131584 094541 905459 33 28 774046 225954 868680 131320 094634 905366 32 29 774217 225783 868945 131055 094728 905272 31 30 9.774388 10.225612 9.869209 10.130791 10.094821 9.905179 30 31 774558 225442 869473 130527 094915 905085 29 32 774729 225271 869737 130263 095008 904992 28 33 774899 225101 870001 129999 095102 904898 27 34 775070 224930 870265 129735 095196 904804 26 35 9.775240 10.224760 9.870529 10.129471 10.095289 9.904711 25 36 775410 224590 870793 129207 095383 904617 24 37 775580 224420 871057 128943 095477 904523 231 38 775750 224250 871321 128679 095571 904429 22 39 775920 224080 871585 128415 095665 904335 21 409.776090 10.223910 19.871849 10.128151 10.095759 9.904241 20 41 776259 223741 872112 127888 095853 904147 191 42 776429 223571 872376 127624 095947 904053 18 43 776598 223402 872640 127360 096041 903959 17 44 776768 223232 872903 127097 096136 903864 16 45 9.776937 10.223063 9.873167 10.126833 10.096230 9.903770 15 46 777106 222894 873430 126570 096324 903676 14 47 777275 222725 873694 126306 096419 903581 13 777444 222556 873957 126043 096513 903487 12 49 777613 222387 874220 125780 096608 903392 11 150 9.777781 10.222219 9.874484 10.125516 10.096702 9.903298 10 51 777950 222050 874747 125253 096797 9032039 52 778119 221881 875010 124990 096892 903108 53 778287 221713 875273 124727 096986 903014 54 778455 221545 875536 124464 097081 9029196 155 9.778624 10.221376 9.875800 10.124200 10.097176 9.902824 778792 221208 876063 123937 097271 902729 57 778960 221040 876326 123674 097366 902634 58 779128 220872 876589 123411 097461 902539 59 779295 220705 876851 123149 097556 902444 60 779463 220537 877114 122886 097651 902349 Cosine. Secant. Cotang. Tang. Cosec. Sine. Page 23
41° Sine. Cosec. Tang. Cotang. Secant. Cosine. 0 9.816943 10.183057 9.939163 10.060837 10.122220 9.877780 60 817088 182912 939418 060582 122330 877670 591 2 817233 182767 939673 060327 122440 877560 158 3 817379 182621 939928 060072 122550 877450 57 4 817524 182476 940183 059817 122660 877340 156 59.817668 10.182332 9.940438 10.059562 10.122770 9.877230 55 6 817813 182187 940694 059306 · 122880 877120 54 7 817958 182042 940949 059051 122990 877010 53 8 818103 181897 941204 058796 123101 876899 52 9 818247 181753 941458 058542 123211 876789 51 109.818392 10.181608 (9.941714 10.058286 10.123322 9.876678 150 11 818536 181464 941968 058032 123432 876568 49 12 818681 181319 942223 057777 123543 876457 48 13 818825 181175 942478 057522 123653 876347 47 14 818969 181031 942733 057267 123764 876236 46 15 9.819113 10.180887 9.942988 10.057012 10.123875 9.876125 45 16 819257 180743 943243 056757 123986 876014 44 17 819401 180599 943498 056502 124096 875904 431 18 819545 180455 943752 056248 124207 875793 42 19 819689 180311 944007 055993 124318 875682 41 2019.819832 10.180168 9.944262 10.055738 10.124429 9.875571 40 21 819976 180024 944517 055483 124541 875459 39 22 820120 179880 944771 055229 124652 875348 381820263 179737 945026 054974 124763 875237 37 24 820406 179594 945281 054719 124874 875126 36 25 9.820550 10.179450 9.945535 10.054465 10.124986 9.875014 35 26 820693 179307 945790 054210 125097 874903 34 27 820836 179164 946045 053955 125209 874791 33 28 820979 179021 946299 053701 125320 874680 32 29 821122 178878 946554 053446 125432 874568 31 309.821265 10.178735 9.946808 10.053192 10.125544 9.874456 30 31 821407 178593 947063 052937 125656 874344 29 32 821550 178450 947318 052682 125768 874232 28 33 821693 178307 947572 052428 125879 874121 27 34 821835 178165 947826 052174 125991 874009 261 35 9.821977 10.178023 9.948081 10.051919 10.126104 9.873896 25 822120 177880 948336 051664 126216 873784 24 822262 177738 948590 051410 126328 873672 23 822404 177596 948844 051156 126440 873560 22 39 822546 177454 949099 050901 126552 873448 21 40 9.822688 10.177312 9.949353 10.050647 10.126665 9.873335 20 41 822830 177170 949607 050393 126777 873223 191 42 822972 177028 949862 050138 126890 873110 18 43 823114 176886 950116 049884 127002 872998 17 823255 176745 950370 049630 127115 872885 16 45 9.823397 10.176603 9.950625 10.049375 10.127228 9.872772 15 46 823539 176461 950879 049121 127341 872659 14 47 823680 176320 951133 048867 127453 872547 13 48 823821 176179 951388 048612 127566 872434 12 49 823963 176037 951642 048358 127679 872321 11 50 9.824104 10.175896 9.951896 10.048104 10.127792 9.872208 1051 824245 175755 952150 047850 127905 8720959 52 824386 175614 952405 047595 128019 871981 853 824527 175473 952659 047341 128132 871868 7 54 824668 175332 952913 047087 128245 871755 6 55 9.824808 10.175192 9.953167 10.046833 10.128359 9.871641 5 56 824949 175051 953421 046579 128472 871528 4 825090 174910 953675 046325 128586 871414 31 Page 24
POINTS AND QUARTER-POINTS REDUCED TO DEGREES, ETC.
0 1 2 48 45 0 1 03 8 26 15 0 3 N. by E. N. by W. 10 11 15 010 $. by E. S. by W. 11 14 3 45 1 1 1 2 16 52 30 1 2 1 3 19 41 15 1 3 N.N.E. N.N.W. 2 0 22 30 0 2 0 || S.S.E. S.S.W. 2 125 18 45 2 1 2 2 28 7 30 2 2 2 3 30 56 15 2 3 N.E. by N. N.W. by N. 3 033 45 0 3 0 S.E. by S. S.W. by S. 31 36 33 45 3 1 3 2 39 22 30 3 2 3 3 42 11 15 3 3 N.E. N.W. 4 045 0 040 S.E. S.W. 4 3 53 26 15 | 4 3 N.E. by E. N.W. by W. 5 0 56 15 0 5 0 S.E. by E. S.W. by W. 5 1 59 3 455 1 5 2 61 52 30 5 2 5 3 64 41 15 5 3 E.N.E. W.N.W. 6 067 30 060 | E.S.E. W.S.W. 6 1 70 18 45 6 1 6 2 73 7 30 6 2 6 3 75 56 15 6 3 E. by N. W. by N. 7 078 45 07 0 | E. by S. W. by S. 7 1 81 33 45 7 1 7 2 84 22 30 7 2 7 3 87 11 15 7 3 East. West. 8 090 00180 East. West. Page 25 |
One of the two acute angles of a right angled triangle is 48 degree find the other acute angle
Pos Terkait
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