Question

If \( \sin \theta = \frac{\sqrt{3}}{2} \), \( 0^\circ<\theta<90^\circ \), then \( \tan^2 \theta - 1 = \) ?

• A. 1
• B. 0
• C. 2
• D. -1

Hint:

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find \( \cos \theta \), then calculate \( \tan^2 \theta - 1 \).

Answer 1

The Correct Option is A

We are given that \( \sin \theta = \frac{\sqrt{3}}{2} \). From the trigonometric identity \( \sin^2 \theta + \cos^2 \theta = 1 \), we can find \( \cos \theta \): \[ \sin^2 \theta + \cos^2 \theta = 1 \quad \Rightarrow \quad \left( \frac{\sqrt{3}}{2} \right)^2 + \cos^2 \theta = 1 \quad \Rightarrow \quad \frac{3}{4} + \cos^2 \theta = 1. \] Solving for \( \cos^2 \theta \): \[ \cos^2 \theta = 1 - \frac{3}{4} = \frac{1}{4} \quad \Rightarrow \quad \cos \theta = \frac{1}{2}. \] Now, \( \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \): \[ \tan^2 \theta = \frac{\left( \frac{\sqrt{3}}{2} \right)^2}{\left( \frac{1}{2} \right)^2} = \frac{\frac{3}{4}}{\frac{1}{4}} = 3. \] Thus, \( \tan^2 \theta - 1 = 3 - 1 = 2 \). The value of \( \tan^2 \theta - 1 \) is \( \boxed{2} \).