Evaluate the expression under the given conditions.

Question

sin(θ + ϕ); sin(θ) = 15/17, θ in Quadrant I, cos(ϕ) = − 5 / 5 , ϕ in Quadrant II

Answer 1

quadrant 1: sine and cosine are both positive

sin T = 15/17 --> opposite is 15 and hypotenuse is 17 ---> adjacent is 8 by pythagorean

cos T = 8/17

quadrant 2: sine is positive and cosine is negative

cos C = -sqrt(5)/5 ---> adjacent is -(Sqrt(5)) and hypotenuse is 5

So the opposite is 2 * sqrt(5) by pythagorean

so sin(C) = (2/5) sqrt(5)

sin (T+c) = sinT cos c + cos T sin c

= (15/17) ( -sqrt(5))/5 + (8/17)(2/5)(sqrt(5))

= (16/85)sqrt(5) - (15/85) sqrt(5)

= (1/5)sqrt(5)