This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...we have cot a = 1.2496+; hence a = 38 40' Tit. When n is odd, the values of a must be discarded. Hence a = 38 40' 2 nir. The reason for the additional values in the last two examples is that in example 2 we really solved both the equations tan 6 sec 9 = 3, and in example 3, both the equations cot a csc a = 2. 165. ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1910 Excerpt: ...we have cot a = 1.2496+; hence a = 38 40' Tit. When n is odd, the values of a must be discarded. Hence a = 38 40' 2 nir. The reason for the additional values in the last two examples is that in example 2 we really solved both the equations tan 6 sec 9 = 3, and in example 3, both the equations cot a csc a = 2. 165. Examples Illustrating Special Methods.--These depend chiefly on transforming the given equation by means of some of the standard formulas. Exercise 1. Check these answers. Draw figures for several values of a as 10, 30, 50, 70. Discuss the case a = 90. Exercise 2. In example 7, in passing from the first equation to the second we divide by tan 5 a, which is permissible only if tan 5a/0, Justify the division. Exercise 3. Justify the division by cos x in example 4. 8. a sin 9 + b cos 9 = c. We might reduce to sin 9 or cos 9 and proceed according to the rule of (164). A method much preferred in practice is as follows. In place of a and b introduce two new constants m and M such that (a-m cos M ( m = Va2 + 6 / 6 = To sin M; ) M = tan-1-v a The given equation then becomes To (sin 9 cos M + cos 9 sin M) = C or sin (S + M) =--TO Hence if we let sin-11 represent all angles whose sine is x, 9+ M = sin-1--, or 9 = sin-1----M. To m 9--sin-1 C--tan-1-Va2 + b2 a Graphic Solution. As an example, we take the equation sin 2 0 + sin 6 + = 0. We want the values of 6 which reduce the expression sin 2 t) + sin0 + 20 zer0 Let y = sin 2 6 + sin0 + j Calculate y for a series of values of 8, as 6 = 0, 10, 20, ..., and plot the points (6, y) in rectangular coordinates. The resulting curve will show the approximate values of 6 for which y is zero. Any convenient scales may be used on the axes of 6 and y. Let the student...
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