Question

If 2 sin θ + 3 cos θ = 2, then 3 sin θ – 2 cos θ = ______.

Answer 1

If 2 sin θ + 3 cos θ = 2, then 3 sin θ – 2 cos θ = ±3.

Explanation:

Square and add the two expressions:

(2 sin θ + 3 cos θ)² + (3 sin θ – 2 cos θ)² = (4 + 9)

(sin² θ + cos² θ) = 13

Given 2 sin θ + 3 cos θ = 2.

Substitute to get 4 + (3 sin θ – 2 cos θ)² = 13.

So (3 sin θ – 2 cos θ)² = 9 and hence 3 sin θ – 2 cos θ = ±3. 

This sign depends on the particular θ.