Question

If a and b are two angles in standard position in Quadrant I, find cos(a+b) for the given function values. sin a=8/17and cos b=12/13

1) -220/221

2) -140/221

3) 140/221

4) 220/221

Answer 1

Recall the sum identity for cosine:

cos(a + b) = cos(a) cos(b) - sin(a) sin(b)

so that

cos(a + b) = 12/13 cos(a) - 8/17 sin(b)

Since both a and b terminate in the first quadrant, we know that both cos(a) and sin(b) are positive. Then using the Pythagorean identity,

cos²(a) + sin²(a) = 1 ⇒ cos(a) = √(1 - sin²(a)) = 15/17

cos²(b) + sin²(b) = 1 ⇒ sin(b) = √(1 - cos²(b)) = 5/13

Then

cos(a + b) = 12/13 • 15/17 - 8/17 • 5/13 = 140/221