If tan θ = 8/7 and cot ϕ = 7/8 , find the exact value of sin ( θ + ϕ ) .

Answer 1

Sketch a triangle in 1st quadrant with adjacent side x = 7, opposite side y = 8. Hypotenuse then = √113. Note that the two angles are the same, since cot = 1/tan. So θ + Φ = 2θ, and sin (θ+Φ) = sin (2θ) = 2 (sin θ)(cos θ) from your trig identities. Then sin (θ+Φ) = 2 (8√113) (7/√113) = 112/113. The angles could be in 3rd quadrant with the same result, because both sin and cos are negative there. If one angle is in 1st and the other is in 3rd quadrant, then the result would be -112/113.