Find the exact value

Question

If tan θ = 3/5 and cot Φ =8/5, find the exact value of sin (θ + Φ).

Answer 1

The angle addition identity for sine says: sin(θ + Φ) = sin(θ)cos(Φ) + cos(θ)sin(Φ) but since we only have values for tan(θ) and cot(Φ), we need to do a little work.

tan(θ) = opposite/adjacent (3/5 in this case) and cot(Φ) is adjacent/opposite (8/5 in this case), so I could draw triangles like this:

Solving for x and y using the Pythagorean Theorem:

x = √(3² + 5²) = √34

y = √(5² + 8²) = √89

So, knowing this information we can write:

sin(θ) = 3/√34

cos(θ) = 5/√34

sin(Φ) = 5/√89

cos(Φ) = 8/√89

I can now plug these into the angle addition identity

sin(θ + Φ) = sin(θ)cos(Φ) + cos(θ)sin(Φ)

sin(θ + Φ) = (3/√34)(8/√89) + (5/√34)(5/√89)

sin(θ + Φ) = 24/√3026 + 25/√3026

sin(θ + Φ) = 49/√3026 = 49√3026/3026