𝜋 < 𝛳 < 2𝜋 and tan𝛳 = 1, find sin𝛳.

Answer 1

We know that sinθ/cosθ = tanθ. If tanθ=1, that means sinθ=cosθ.

Recall the unit circle. 𝜋<θ<2𝜋 in the 3rd and 4th quadrants, where y is negative. For which angle θ in this range does cosθ=sinθ? What is the sin of that angle?

Answer: cosθ=sinθ at the angle 𝜋+𝜋/4 = 5𝜋/4. We then know that sinθ = sin(5𝜋/4) = -1/√2.