Trigonometry & Inverse Trigonometry
Trigonometric Equations and Identities
Grade 11
Question:
<p>If <span>√</span>2 cos A = cos B + cos³ B, and <span>√</span>2 sin A = sin B - sin³ B then sin(A - B) = ?</p>
<p>(a) ±1</p>
<p>(b) ±1/2</p>
<p>(c) ±1/(2√2)</p>
<p>(d) ±1/4</p>
Step-by-Step Solution
Key Concept: Use the given trigonometric identities and manipulate them algebraically to express sin(A - B) in terms of B, then apply the constraint from squaring and adding both equations.
<p><strong>Step 1:</strong> Given: <span>√</span>2 cos A = cos B + cos³ B ... (i)</p><p>and <span>√</span>2 sin A = sin B - sin³ B ... (ii)</p><p><strong>Step 2:</strong> Calculate sin(A - B):</p><p><span>√</span>2 sin A cos B - <span>√</span>2 cos A sin B = (sin B - sin³ B) cos B - (cos B + cos³ B) sin B</p><p>= -sin B cos B</p><p><strong>Step 3:</strong> Therefore, sin(A - B) = -sin B cos B / <span>√</span>2</p><p><strong>Step 4:</strong> Squaring and adding equations (i) and (ii):</p><p>2 = cos² B + sin² B + cos⁶ B + sin⁶ B + 2(cos⁴ B - sin⁴ B)</p><p>1 = 1 - (3/4) sin² 2B + 2 cos 2B</p><p>∴ Answer is (c) ±1/(2√2)</p>
Correct Answer: C