Question

For positive acute angles A and B, it is known that cos A = 4 and sin B = HFind the value of cos(A - B) in simplest form.

Answer 1

cos ( A-B) = cos A cos B + sin A sin B

Cos A= 4/5 and Sin B= 15/17

Cos A = 4/5

CosA = adj / hyp = 4/5

adj ^2 + opp ^2 = hyp ^2

4^2 + opp ^2 = 5^2

opp ^2 = 25 - 16

opp^2 = 9

opp = 3

Since they are positive acute angles, it is in the first quadrant and will be positive

Sin A = opp / hyp = 3/5

Sin B = 15/17 = opp / hyp

opp^2 + adj ^2 = hyp ^2

15^2 + adj ^2 = 17^2

225 + adj^2 =289

adj ^2 =64

adj = 8

Since they are positive acute angles, it is in the first quadrant and will be positive

Cos B = adj / hyp = 8/17

cos ( A-B) = cos A cos B + sin A sin B = 4/5 * 8/17 + 3/5 * 15/17 = 32/85 + 45/85 = 77/85