Question

If \( 0 <\theta <\frac{\pi}{4} \) and \( 8\cos\theta + 15\sin\theta = 15 \), then \( 15\cos\theta - 8\sin\theta = \)

• A. \( 15 \)
• B. \( 7 \)
• C. \( 8 \)
• D. \( 23 \)

Hint:

Using trigonometric identities to transform expressions simplifies calculations. Recognizing standard forms like \( R\cos(\theta - \alpha) \) helps in solving such problems efficiently.

Answer 1

The Correct Option is C

Step 1: Express the Given Equation in Standard Form 
We are given: \[ 8\cos\theta + 15\sin\theta = 15. \] Comparing with the standard form \( R\cos(\theta - \alpha) \), we first determine \( R \): \[ R = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17. \] 

Step 2: Express Trigonometric Components 
Let: \[ 8 = 17\cos\alpha, \quad 15 = 17\sin\alpha. \] Dividing both by 17, \[ \cos\alpha = \frac{8}{17}, \quad \sin\alpha = \frac{15}{17}. \] Rewriting the equation: \[ 17 \left( \cos\alpha\cos\theta + \sin\alpha\sin\theta \right) = 15. \] Using the cosine addition formula: \[ \cos(\theta - \alpha) = \frac{15}{17}. \] 

Step 3: Finding the Required Expression 
We need to evaluate: \[ 15\cos\theta - 8\sin\theta. \] Using the same transformation: \[ 15\cos\theta - 8\sin\theta = 17 (\cos\theta \cos\beta - \sin\theta \sin\beta), \] where \( \cos\beta = \frac{15}{17} \) and \( \sin\beta = \frac{8}{17} \), giving: \[ \cos(\theta + \beta) = \frac{8}{17}. \] Multiplying by 17: \[ 15\cos\theta - 8\sin\theta = 8. \]