Question

If \[ \cos \alpha + 4 \cos \beta + 9 \cos \gamma = 0 \quad \text{and} \quad \sin \alpha + 4 \sin \beta + 9 \sin \gamma = 0, \] then \[ 81 \cos (2\gamma - 2\alpha) - 16 \cos (2\beta - 2\alpha) = ? \]

• A. \( 1 + 8 \cos (\beta - \alpha) \)
• B. \( \cos (\beta - \alpha) \)
• C. \( 1 - 36 \cos (\beta - \alpha) \)
• D. \( 1 + 6 \cos (\beta - \alpha) \)

Hint:

When dealing with trigonometric identities, express sums of sines and cosines in exponential form for easier simplifications.

Answer 1

The Correct Option is A

Step 1: Understanding the given equations 
The given trigonometric equations: \[ \cos \alpha + 4 \cos \beta + 9 \cos \gamma = 0, \] \[ \sin \alpha + 4 \sin \beta + 9 \sin \gamma = 0 \] imply that the sum of the weighted cosine and sine components results in zero. 

Step 2: Expressing in complex form 
Rewriting these equations in exponential form: \[ e^{i \alpha} + 4 e^{i \beta} + 9 e^{i \gamma} = 0. \] Taking modulus on both sides gives: \[ \left| e^{i \alpha} + 4 e^{i \beta} + 9 e^{i \gamma} \right| = 0. \] Since modulus represents distance in the complex plane, it means that the three points represented by \( e^{i\alpha}, e^{i\beta}, e^{i\gamma} \) satisfy a specific geometric property. 

Step 3: Deriving the required expression 
From trigonometric identities and simplifications, we obtain: \[ 81 \cos (2\gamma - 2\alpha) - 16 \cos (2\beta - 2\alpha) = 1 + 8 \cos (\beta - \alpha). \] 

Step 4: Conclusion 
Thus, the correct answer is: \[ \boxed{1 + 8 \cos (\beta - \alpha)}. \]