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:

When solving trigonometric equations, use identities and the quadratic formula to simplify the expressions and solve for the trigonometric function.

Answer 1

The Correct Option is C

Step 1: Start with the given equation: \[ 12\cot^2 \theta - 31\csc \theta + 32 = 0. \] Using the identity \( \cot^2 \theta = \csc^2 \theta - 1 \), substitute into the equation: \[ 12(\csc^2 \theta - 1) - 31\csc \theta + 32 = 0. \] Simplify: \[ 12\csc^2 \theta - 12 - 31\csc \theta + 32 = 0 \quad \Rightarrow \quad 12\csc^2 \theta - 31\csc \theta + 20 = 0. \] Let \( x = \csc \theta \), so the equation becomes: \[ 12x^2 - 31x + 20 = 0. \] Solve this quadratic equation using the quadratic formula: \[ x = \frac{-(-31) \pm \sqrt{(-31)^2 - 4(12)(20)}}{2(12)} = \frac{31 \pm \sqrt{961 - 960}}{24} = \frac{31 \pm 1}{24}. \] Thus, \( x = \frac{32}{24} = \frac{4}{3} \) or \( x = \frac{30}{24} = \frac{5}{4} \). 
Step 2: Since \( \csc \theta = \frac{1}{\sin \theta} \), we have: \[ \sin \theta = \frac{3}{4} \quad {or} \quad \sin \theta = \frac{4}{5}. \]