By William J. Bruce, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)
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Extra resources for Analytic Trigonometry
120°, 120°, 240° 42 ANALYTIC TRIGONOMETRY FIG. 22 Example 2. 61160. 61160 lie in quadrants I and III (Fig. 23). 61160 Thus from tables a = 31° 27' and we get Θ = . . , — 148° 33', 31° 27', 211° 27' FIG. 23 TRIGONOMETRIC FUNCTIONS OF ANGLES 43 Suppose now that we want the set of all angles which satisfy the equation sin Θ cos Θ — \ sin Θ = 0. This is similar to the algebraic equation xy — % x — 0 and to solve the latter we would write x ( y — £) = 0 and obtain x = 0 and y = \ as solutions. Applying our algebra to the trigonometric equation, we get and hence, all values of 0 are such that sin Θ = 0 and cos Θ 0, or cos Θ Up to this point, there is a precise similarity with the solution of the algebraic equation but, since we are not just concerned with finding possible values for sin 0 and cos Θ, our problem is merely reduced to solving two simple equations for Θ.
8. sin 3a cos a — cos 3a sin a = sin 2a. FUNCTIONS OF SUMS AND RELATED FUNCTIONS tan 2a + tan a r-— = tan 3a 1 — tan 2a tan a 10. tan 2 0 - tan 0 = tan 0 sec 2 0. Λ 9. t 11. csc 0 - cot 0 -tan ( 4 ) sin 2 0 12. — — = tan 0. 1 + cos 2 0 13. sec 2 0 1 - 2 sin2 0 * 14. (sin 0 - cos 0)2 = 1 - sin 2 0. 15. tan 2 0 cot 0 16. 1 - sin 2 0 cos 2 0 2 1 - tan2 0 1 - tan 0 1 + tan 0 ' Solve the following equations for 0 < 0 < 2 π: 17. sin 2 0 = i λ/3" 18. cos 2 0 = - -^- . 19. 1 - 2 sin2 2 0 = h (Show two methods).
20 TRIGONOMETRIC FUNCTIONS OF ANGLES 37 From the reference triangle, we get sin Θ = -f-, tan Θ = — f, cot Θ = — Î, sec Θ = — | , and csc 0 = f. Example 2. Find the other functions of Θ if sin 0 = — %. Since the quadrant of Θ is not specified, we must consider Θ in quadrants HI and IV because sin Θ is negative. But sin θ = y r and, since r is positive, we choose r = 3 and get >> = — 1. It follows that x = ± V8 and two reference triangles are needed V8 (Fig. 20). From the reference triangles, we get cos Θ = ± — , tan Θ = ± —-_ , cot Θ = ± V8, sec 0 = ± —=.