Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>If <span style='font-family: Arial'>a cos³ α + 3a cos α sin² α = m</span> and <span style='font-family: Arial'>a sin³ α + 3a cos² α sin α = n</span>, then <span style='font-family: Arial'>(m + n)^(2/3) + (m - n)^(2/3)</span> is equal to</p>
<p>(a) <span style='font-family: Arial'>2a</span></p>
<p>(b) <span style='font-family: Arial'>2a^(1/3)</span></p>
<p>(c) <span style='font-family: Arial'>2a^(2/3)</span></p>
<p>(d) <span style='font-family: Arial'>2a^3</span></p>
Step-by-Step Solution
Key Concept: Factor expressions and recognize patterns to simplify the fractional power expressions.
<p><strong>Step 1:</strong> Factor out from given expressions: <span style='font-family: Arial'>m = a cos α(cos² α + 3 sin² α) = a cos α(1 + 2 sin² α)</span></p><p><strong>Step 2:</strong> Similarly, <span style='font-family: Arial'>n = a sin α(sin² α + 3 cos² α) = a sin α(1 + 2 cos² α)</span></p><p><strong>Step 3:</strong> Note that <span style='font-family: Arial'>m + n = a(cos α + sin α) · (1 + 2 sin α cos α)</span> and compute accordingly</p><p><strong>Step 4:</strong> Using algebraic identities and simplification: <span style='font-family: Arial'>(m + n)^(2/3) + (m - n)^(2/3) = 2a^(2/3)</span></p><p>∴ Answer is C.</p>
Correct Answer: C