Question

The range of \( 2 \left| \sin x + \cos x \right| - \sqrt{2} \) is:

• A. \( \left[ -\sqrt{2}, \sqrt{2} \right] \)
• B. \( \left[ -3\sqrt{2}, \sqrt{2} \right] \)
• C. \( \left( -3\sqrt{2}, \sqrt{2} \right) \)
• D. None of these

Hint:

To determine the range of functions involving trigonometric identities, transform the expressions into standard forms and consider the range of trigonometric functions such as sine and cosine.

Answer 1

The Correct Option is A

We start by finding the range of \( \sin x + \cos x \). Step 1: Express in an alternate form \[ \sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \] Since the sine function satisfies \( -1 \leq \sin \theta \leq 1 \), we obtain: \[ -\sqrt{2} \leq \sin x + \cos x \leq \sqrt{2} \] Taking the modulus: \[ 0 \leq \left| \sin x + \cos x \right| \leq \sqrt{2} \] Multiplying both sides by 2: \[ 0 \leq 2 \left| \sin x + \cos x \right| \leq 2\sqrt{2} \]
Step 2: Adjusting for the given function \[ -\sqrt{2} \leq 2 \left| \sin x + \cos x \right| - \sqrt{2} \leq \sqrt{2} \] Final Answer: \[ \boxed{\left[ -\sqrt{2}, \sqrt{2} \right]} \]