Trigonometric Ratios and Identities
Compound Angles
AIEEE 2004
Grade 11
Question:
Let $\sin \alpha + \sin \beta = -\frac{21}{65}$ and $\cos \alpha + \cos \beta = -\frac{27}{65}$, where $\pi < \alpha - \beta < 3\pi$. Then $\cos \frac{\alpha - \beta}{2}$ equals:
(1) $-\frac{6}{65}$
(2) $\frac{3}{\sqrt{130}}$
(3) $-\frac{3}{\sqrt{130}}$
(4) $\frac{6}{65}$
Step-by-Step Solution
Key Concept: Square and add both equations: $(\cdot)^2 + (\cdot)^2 = 2 + 2 \cos(\alpha - \beta)$. Solve for $\cos(\alpha - \beta)$, then use $2 \cos^2 \frac{\alpha - \beta}{2} = 1 + \cos(\alpha - \beta)$. Check the sign from the given range of $\alpha - \beta$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)