Question

If a sinθ + b cosθ = p and a cosθ – b sinθ = q, which of the following equation is correct?

(A) a + b² = p² + q²
(B) a² + b² = p² + q²
(C) a² - b² = p² - q²
(D) a - b² = p - q²

Answer 1

To find the correct equation, we square and add the given equations, using the identity (sinθ)² + (cosθ)² = 1, which leads to the correct equation: a² + b² = p² + q².

Step-by-step explanation:

Given the equations:

• a sinθ + b cosθ = p
• a cosθ − b sinθ = q

We need to find which of the following equations is correct.

Squaring both the given equations separately and adding them will eliminate the sinθ and cosθ terms, as (sinθ)² + (cosθ)² = 1:

1. (a sinθ + b cosθ)² = p²
2. (a cosθ − b sinθ)² = q²

Expanding these and adding them:

(a² sin²θ + 2ab sinθ cosθ + b² cos²θ) + (a² cos²θ - 2ab sinθ cosθ + b² sin²θ) = p² + q².

Simplifying:

a²(sin²θ + cos²θ) + b²(sin²θ + cos²θ) = p² + q².

Since sin²θ + cos²θ = 1, we have:

a² + b² = p² + q².

Thus, the correct equation is:

• (B) a² + b² = p² + q²