If $ \sin \theta + \cos \theta = 1 $, find the value of $ \sin \theta \cos \theta $.
Show Hint
- \( 0 \)
- \( \frac{1}{2} \)
- \( \frac{1}{4} \)
- \( \frac{3}{4} \)
The Correct Option is A
Solution and Explanation
Given: \[ \sin \theta + \cos \theta = 1 \] Step 1: Square both sides \[ (\sin \theta + \cos \theta)^2 = 1^2 \] Expanding the left side using the identity \((a+b)^2 = a^2 + 2ab + b^2\), we get: \[ \sin^2 \theta + 2 \sin \theta \cos \theta + \cos^2 \theta = 1 \] Step 2: Use the Pythagorean identity Recall that: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substitute this into the equation: \[ 1 + 2 \sin \theta \cos \theta = 1 \] Step 3: Simplify Subtract 1 from both sides: \[ 2 \sin \theta \cos \theta = 0 \] Divide both sides by 2: \[ \sin \theta \cos \theta = 0 \] Step 4: Verify possible values - If \(\sin \theta = 0\), then \(\cos \theta = 1\), and \(\sin \theta \cos \theta = 0 \times 1 = 0\). - If \(\cos \theta = 0\), then \(\sin \theta = 1\), and \(\sin \theta \cos \theta = 1 \times 0 = 0\). Both cases satisfy the equation.
Final answer: \[ \boxed{0} \]
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