Find the exact value for the expression under the given conditions. (cosα+β) for sinα = −3/5 for α in Quadrant III and cosβ = −3/4 for β in Quadrant II.

Answer 1

The identity for: cos (α + β) = (cos α)(cos β) - (sin α)(sin β)

sin α = (-3/5) and cos b = (-3,/4) are given, thus cos α and sin β need to be determined.

Using (cos α)^2 + (sin α)^2 = 1 and (cos β)^2 + (sin β)^2 = 1

(cos α)^2 + (-3/5)^2 = 1 (-3/4)^2 + (sin β)^2 = 1

(cos α)^2 + (9/25) = 1 (9/16) + (sin β)^2 = 1

(cos α)^2 = (25/25) - (9/25) (sin β)^2 = (16/16) - (9/16)

(cos α)^2 = 16/25 (sin β)^2 = (7/16)

cos α = ±(4/5) (sin β) = ±(√7/4)

α is quadrant III β is in Quadrant II

cos α = (-4/5) in Quadrant III sin β = (√7/4) in quadrant II

Thus cos (α + β) = (cos α)(cos β) - (sin α)(sin β)

= (-4/5)(-3/4) - (√7/4)(-3/5)

= (12/20) + (3√7/20)

= 12 +3√7

20