Let sin a=3/5, with a in Quadrant II, and let cos B=5/13, with B in Quadrant IV. Find the exact values for each of the following. a) sin (a+B) b) cos (a-B) c) tan 2a d) sin B/2

Answer 1

With a sketch and Pythagorean Theorem we can fill in the other main function values: cos a = -4/5 and sin B = -12/13. Then use various identities for the following:

a) sin(a+B) = sin a cos B + cos a sin B = (3/5)(5/13) + (-4/5)(-12/13) = (15 - 48)/65 = -33/65.

b) cos(a-B) = cos a cos B + sin a sin B = (-4/5)(5/13) + (3/5)(-12/13) = (-20 -36)/65 = -56/65.

c) tan 2a = 2 tan a /(1 - tan² a) = 2(-3/4) /(1 - 9/16) = (-3/2) /(7/16) = (-3/2) • (16/7) = -24/7.

d) sin B/2 = ±[(1 - cos B)/2]^1/2 = ±[(1 - 5/13)/2]^1/2 = ±[(8/13)/2 = 4/13]^1/2 = ±(2√13)/13.