Question

If \( \sin \theta = \sqrt{2} \cos \theta \), then the value of \( \sec \theta \) is:

• A. \( \frac{1}{\sqrt{3}} \)
• B. \( \sqrt{3} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( \frac{2}{\sqrt{3}} \)

Hint:

Use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to find the value of \( \sec \theta \) when given a relationship between \( \sin \theta \) and \( \cos \theta \).

Answer 1

The Correct Option is B

We are given that \( \sin \theta = \sqrt{2} \cos \theta \). To find \( \sec \theta \), we use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = \sqrt{2} \cos \theta \) into the identity: \[ (\sqrt{2} \cos \theta)^2 + \cos^2 \theta = 1. \] Simplify: \[ 2 \cos^2 \theta + \cos^2 \theta = 1 \quad \Rightarrow \quad 3 \cos^2 \theta = 1 \quad \Rightarrow \quad \cos^2 \theta = \frac{1}{3}. \] Thus: \[ \cos \theta = \frac{1}{\sqrt{3}}. \] Now, \( \sec \theta = \frac{1}{\cos \theta} = \sqrt{3} \). Thus, the value of \( \sec \theta \) is \( \boxed{\sqrt{3}} \).