Question

If \( 12 \cot^2 \theta - 31 \csc \theta + 32 = 0 \), then the value of \( \sin \theta \) is:

• A. \( \frac{3}{5} \) or 1
• B. \( \frac{2}{3} \) or \( -\frac{2}{3} \)
• C. \( \frac{4}{5} \) or \( \frac{3}{4} \)
• D. \( \pm \frac{1}{2} \)

Hint:

To solve equations involving trigonometric identities, first express all terms in terms of \( \sin \theta \) or \( \cos \theta \), and then solve the resulting algebraic equation.

Answer 1

The Correct Option is A

Step 1: Start with the given equation: \[ 12 \cot^2 \theta - 31 \csc \theta + 32 = 0. \] Use the identity \( \cot^2 \theta = \csc^2 \theta - 1 \). Substituting gives: \[ 12(\csc^2 \theta - 1) - 31 \csc \theta + 32 = 0. \] Step 2: Solve the quadratic equation for \( \csc \theta \), and use the identity \( \csc \theta = \frac{1}{\sin \theta} \) to find the possible values of \( \sin \theta \). The solution gives \( \sin \theta = \frac{3}{5} \) or 1.

Final Answer: \[ \boxed{\frac{3}{5} \text{ or } 1} \]