This makes the limits for sin closer than in (1) of this Art. 135. To calculate the Sine and Cosine of 10" and of 1'. (1) Let be the circular measure of 10". or = 4 0 = .000048481368110 ......., correct to 15 decimal places. 03 .000000000000032 ....., 66 =.000048481368078 ..., 66 66 66 4 .. sin 10".000048481368, to 12 decimal places. = .9999999988248..., to 13 decimal places. Or we may use the results established in (2) of Art. 134, and obtain the same value. .. sin 1'= .00029088820 to 11 decimal places. cos 1'=√1-sin2 1'=.999999957692025 to 15 decimal places, .. cos 1'= .999999957692025 to 15 decimal places, as before. Cor. 1. The sine of 10" equals the circular measure of 10", to 12 decimal places; and the sine of 1' equals the circular measure of 1' to 11 decimal places. Cor. 2. If n denote any number of seconds less than 60, we shall have approximately for the sine of n" sin n"= n sin 1", the circular measure of n", approxi mately, =n times the circular measure of 1". Cor. 3. n= circular measure of n" sin 1" approximately; that is, the number of seconds in any small angle is found approximately by dividing the circular measure of that angle by the sine of one second. 136. To construct a Table of Natural Sines and Cosines at Intervals of 1'. We have, by Art. 45, sin(x+y)=2 sin x cos y sin (x − y), cos(x + y) = 2 cos x cos y — cos (x − y). Suppose the angles to increase by 1'; putting y = 1', we have, sin(x+1)=2 sin x cos 1'-sin(x — 1'). = Putting a 1', 2', 3', 4', etc., in (1) and (2), we get for the sines sin 2' = 2 sin 1' cos 1' - sin 0' = .0005817764, sin 3' 2 sin 2' cos 1' — sin 1' = .0008726646, sin 4' = 2 sin 3' cos 1'- sin 2' = .0011635526; and for the cosines = cos 2' 2 cos 1' cos 1' - cos 0' = .9999998308, cos 4' 2 cos 3' cos 1' - cos 2' = .9999993223. = We can proceed in this manner* until we find the values of the sines and cosines of all angles at intervals of 1' from 0° to 30°. 137. Another Method. Let a denote any angle. Then, in the identity, sin(n+1)-sin na=sin na-sin(n−1)α-k sin na . (1) * This method is due to Thomas Simpson, an English geometrician. This formula enables us to construct a table of sines of angles whose common difference is a. Thus, suppose α = 10", and let n = 1, 2, 3, 4, etc. sin 30" sin 30" . sin 20" -k sin 30", etc. These equations give in succession sin 20", sin 30", etc. It will be seen that the most laborious part of this work is the multiplication of k by the sines of 10", 20", etc., as they are successively found. But from the value of cos 10", we have the smallness of which facilitates the process. In the same manner a table of cosines can be constructed by means of the formula, 138. The Sines and Cosines from 30° to 60°. — It is not necessary to calculate in this way the sines and cosines of angles beyond 30°, as we can obtain their values for angles from 30° to 60° more easily by means of the formulæ (Art. 45): sin (30° + α): = cos α sin (30° — α), cos (30° + α) = cos (30° — α) — sin α, by giving a all values up to 30°. Thus, sin 30° 1' = cos 1'- sin 29° 59', cos 30° 1' = cos 29° 59′ — sin 1', and so on. |