Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 26) SINE AND COSINE FUNCTIONS 197Hence, since 135" is in the second quadrant,8sin 135" = + = $(d) 17" cannot be expressed in terms of more familiar angles (such as 30", 45", 60") in such a way as toallow the application of any of our formulas. We must then use the cosine table in Appendix D, whichgives 0.9563 as the value of cos 17". This is an approximation, correct to four decimal places.26.2 Prove Theorem 26.4.Refer to Fig. 26-9. Let D be the foot of the perpendicular from A to the x-axis. Let F be the point onthe ray OA at a unit distance from the origin. Then, F = (cos 8, sin 8). If E is the foot of the perpendicularfrom F to the x-axis, we have- -OE = cos e FE = sin 8Now, AADO is similar to AFEO (by the AA criterion), whenceTherefore, x = r cos 8 and y = r sin 8. When A(x, y) is in one of the other quadrants, the proof can bereduced to the case where A(x, y) is in the first quadrant. The proof is easy when A(x, y) is on the x-axis oron the y-axis.E D XFig. 26-926.3 Prove Theorem 26.5.In Fig. 26-7, set up a coordinate system with C as origin and B on the positive x-axis. Then B hascoordinates (a, 0). Let (x, y) be the coordinates of A. By Theorem 26.4,= b cos e y = b sin 8By the distance formula (24,Hence,c = J(x- a)' + (y - 0)' = & qTj7c2 = (x - a)2 + y2 = (b cos 8 - a)' + (b sin q2ALGEBRA(P - Q)' = P' - 2PQ + Q'= b2 cos2 8 - h b COS e + a2 + b2 sin' 8= U' + b2(cos2 8 + sin' 8) - 2ab COS 8= a2 + b2 - 2ab cos e