<p>Express the following in <em>a + ib</em> form:<br>(a) \(\dfrac{(\cos\alpha + i\sin\alpha)^4}{(\sin\beta + i\cos\beta)^5}\)</p>
Step-by-Step Solution
Key Concept: Convert trigonometric forms to exponential form using Euler's formula, then simplify using exponent rules. Recognize that sin(β) + i·cos(β) = -i(cos(β) + i·sin(β)), which introduces an extra factor of (-i)^5 in the denominator.
<p><strong>Step 1:</strong> Express numerator using Euler's formula: (cos α + i sin α)⁴ = e^(i·4α)</p><p><strong>Step 2:</strong> Rewrite denominator: sin β + i cos β = i(cos β - i sin β) = i·e^(-iβ) = e^(iπ/2)·e^(-iβ) = e^(i(π/2 - β))</p><p><strong>Step 3:</strong> Therefore (sin β + i cos β)⁵ = e^(i·5(π/2 - β)) = e^(i(5π/2 - 5β))</p><p><strong>Step 4:</strong> Compute the fraction: e^(i·4α) / e^(i(5π/2 - 5β)) = e^(i(4α - 5π/2 + 5β)) = e^(i(4α + 5β - 5π/2))</p><p><strong>Step 5:</strong> Since e^(-i·5π/2) = e^(-i·π/2) = -i, we have: e^(i(4α + 5β))·(-i) = (cos(4α + 5β) + i sin(4α + 5β))·(-i)</p><p><strong>Step 6:</strong> Multiply: -i·cos(4α + 5β) - i²·sin(4α + 5β) = sin(4α + 5β) - i cos(4α + 5β)</p><p>∴ Answer: <strong>sin(4α + 5β) - i cos(4α + 5β)</strong></p>
Correct Answer: sin(4α + 5β) - i cos(4α + 5β)