<p>Express the following in <em>a + ib</em> form:<br>(c) \(\dfrac{(\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)}{(\cos\gamma + i\sin\gamma)(\cos\delta + i\sin\delta)}\)</p>
Step-by-Step Solution
Key Concept: Use Euler's form: cos θ + i sin θ = e^(iθ), so multiplication becomes exponent addition and division becomes exponent subtraction. The final result follows from e^(i(α+β-γ-δ)) = cos(α+β-γ-δ) + i sin(α+β-γ-δ).
<p><strong>Step 1:</strong> Express each complex number using De Moivre's form (or Euler's notation).</p><p>cos α + i sin α = e^(iα), cos β + i sin β = e^(iβ), cos γ + i sin γ = e^(iγ), cos δ + i sin δ = e^(iδ)</p><p><strong>Step 2:</strong> Apply the division rule for complex numbers in exponential form.</p><p>The given expression becomes: e^(iα) · e^(iβ) / (e^(iγ) · e^(iδ)) = e^(i(α+β)) / e^(i(γ+δ)) = e^(i(α+β-γ-δ))</p><p><strong>Step 3:</strong> Convert back to trigonometric form using e^(iθ) = cos θ + i sin θ.</p><p>e^(i(α+β-γ-δ)) = cos(α + β - γ - δ) + i sin(α + β - γ - δ)</p><p>∴ Answer: <strong>cos(α + β - γ - δ) + i sin(α + β - γ - δ)</strong></p>
Correct Answer: cos(α + β - γ - δ) + i sin(α + β - γ - δ)