Question

If \( \sin \theta + \cos \theta = \sqrt{2} \), what is the value of \( \sin \theta \cos \theta \)?

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{4} \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

For problems involving trigonometric identities, remember to use the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to simplify expressions and solve for unknown values.

Answer 1

The Correct Option is A

Step 1: We are given that \( \sin \theta + \cos \theta = \sqrt{2} \). Squaring both sides, we get: \[ (\sin \theta + \cos \theta)^2 = (\sqrt{2})^2 \] \[ \sin^2 \theta + 2\sin \theta \cos \theta + \cos^2 \theta = 2 \] Since \( \sin^2 \theta + \cos^2 \theta = 1 \) (Pythagorean identity), we can substitute this into the equation: \[ 1 + 2\sin \theta \cos \theta = 2 \] \[ 2\sin \theta \cos \theta = 1 \] \[ \sin \theta \cos \theta = \frac{1}{2} \]