Question

If \( \sin 2x = 4 \cos x \), then \( x \) is equal to:

• A. \( \frac{n\pi}{2}, \, n \in \mathbb{Z} \)
• B. no value
• C. \( n\pi + (-1)^n \frac{\pi}{4}, \, n \in \mathbb{Z} \)
• D. \( 2n\pi + \frac{\pi}{2}, \, n \in \mathbb{Z} \)

Hint:

When solving trigonometric equations, first check for values that make denominators zero, and then use known identities to simplify the equation.

Answer 1

The Correct Option is D

We are given the equation: \[ \sin 2x = 4 \cos x \] Step 1: Use the double angle identity Using the double angle identity \( \sin 2x = 2 \sin x \cos x \), we can rewrite the equation as: \[ 2 \sin x \cos x = 4 \cos x \] Step 2: Solve the equation If \( \cos x \neq 0 \), we can divide both sides by \( \cos x \): \[ 2 \sin x = 4 \quad \Rightarrow \quad \sin x = 2 \] This is impossible since the sine of an angle can never exceed 1. Therefore, \( \cos x = 0 \). Step 3: Find the solution for \( x \) When \( \cos x = 0 \), \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. Therefore, the solution is: \[ x = 2n\pi + \frac{\pi}{2}, \quad n \in \mathbb{Z} \] Thus, the correct answer is \( 2n\pi + \frac{\pi}{2}, \, n \in \mathbb{Z} \).