Question

If \( a \) is in the 3rd quadrant, \( \beta \) is in the 2nd quadrant such that \( \tan \alpha = \frac{1}{7}, \sin \beta = \frac{1}{\sqrt{10}} \), then \[ \sin(2\alpha + \beta) = \]

• A. \( \frac{3 \times \sqrt{10}}{25} \)
• B. \( \frac{3}{\sqrt{10}} \)
• C. \( \frac{3}{25\sqrt{10}} \)
• D. \( \frac{\sqrt{10}}{3 \times 25} \)

Hint:

For trigonometric expressions involving multiple angles, use the angle addition and double angle formulas to express the function in terms of simpler trigonometric functions. Then simplify using known identities.

Answer 1

The Correct Option is C

Step 1: Find \( \sin(\alpha) \) and \( \cos(\alpha) \). Given \( \tan(\alpha) = \frac{1}{7} \), we calculate \( \sin(\alpha) \) and \( \cos(\alpha) \) using the identity \( \tan^2(\alpha) + 1 = \sec^2(\alpha) \). 
Step 2: Find \( \sin(\beta) \) and \( \cos(\beta) \). Given \( \sin(\beta) = \frac{1}{\sqrt{10}} \), we calculate \( \cos(\beta) \) using the identity \( \sin^2(\beta) + \cos^2(\beta) = 1 \). 
Step 3: Apply the angle addition formula for \( \sin(2\alpha + \beta) \). We use the identity \( \sin(2\alpha + \beta) = \sin(2\alpha) \cos(\beta) + \cos(2\alpha) \sin(\beta) \), and the double angle formulas for sine and cosine to calculate the value of \( \sin(2\alpha + \beta) \). 
Step 4: Final result. The final result is: \[ \sin(2\alpha + \beta) = \frac{3 \times \sqrt{10}}{25}. \]