Question

If \( \sin(\theta) = \frac{15}{17} \), then for \( 0^\circ < \theta < 90^\circ \), \[ \frac{15 \cot(\theta) + 17 \sin(\theta)}{8 \tan(\theta) + 16 \sec(\theta)} \]

• A. \( \frac{23}{49} \)
• B. \( \frac{22}{49} \)
• C. \( \frac{18}{49} \)
• D. \( \frac{17}{49} \)

Hint:

Use the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find unknown trigonometric values and simplify expressions.

Answer 1

The Correct Option is A

Given \( \sin(\theta) = \frac{15}{17} \), we can use the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) to find \( \cos(\theta) \): \[ \cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{15}{17}\right)^2} = \frac{8}{17}. \] Now, we can use the values of \( \sin(\theta) \) and \( \cos(\theta) \) to calculate the trigonometric functions in the given expression and simplify to get \( \frac{23}{49} \).