Consider the complex number \(z = \cos\theta + i\sin\theta\).
(a) Use De Moivre’s theorem to show that \(z^n + \frac{1}{z^n} = 2\cos(n\theta)\) for \(n \in \mathbb{Z}^+\). [3]
(b) Expand \((z + \frac{1}{z})^4\) using the binomial theorem. [3]
(c) Hence, show that \(\cos^4\theta = \frac{1}{8}(\cos(4\theta) + 4\cos(2\theta) + 3)\). [6]
(d) Find the value of \(\int_{0}^{\pi/2} \cos^4\theta\,d\theta\). [5]
Let \(w = \cos\alpha + i\sin\alpha\). Consider the geometric transformation \(T\) in the complex plane defined by \(T(u) = wu\), where \(u\) is a complex number.
(e) Describe the geometric transformation \(T\). [2]
Consider the points A, B, C in the complex plane represented by the complex numbers \(1\), \(1+i\), and \(i\) respectively. Let A’, B’, C’ be the images of A, B, C under the transformation \(T\) with \(w = \cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})\).
(f) Find the complex numbers representing A’, B’, and C’. [4]
(g) Show that triangle A’B’C’ is congruent to triangle ABC. [5]
Question
Use De Moivre’s theorem to show that \(z^n + \frac{1}{z^n} = 2\cos(n\theta)\) for \(n \in \mathbb{Z}^+\). [3]
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Express \(z\) in polar form and apply De Moivre’s theorem to both \(z^n\) and its reciprocal.
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By writing \(z^n = \cos(n\theta) + i\sin(n\theta)\) and \(1/z^n = \cos(n\theta) – i\sin(n\theta)\), their sum equals \(2\cos(n\theta)\).
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Expand \((z + \frac{1}{z})^4\) using the binomial theorem. [3]
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Use the binomial expansion formula \((a+b)^4\) with \(a=z\) and \(b=1/z\).
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The expansion yields \(z^4 + 4z^2 + 6 + 4z^{-2} + z^{-4}\).
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Hence, show that \(\cos^4\theta = \frac{1}{8}(\cos(4\theta) + 4\cos(2\theta) + 3)\). [6]
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Recognize that \(z+1/z = 2\cos\theta\) and equate \((2\cos\theta)^4\) with the expansion from part (b).
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Since \((z+1/z)^4 = 16\cos^4\theta\) and also equals \(z^4+4z^2+6+4z^{-2}+z^{-4}\), using the identities \(z^4+z^{-4}=2\cos(4\theta)\) and \(z^2+z^{-2}=2\cos(2\theta)\), we deduce \(\cos^4\theta = \frac{1}{8}(\cos(4\theta)+4\cos(2\theta)+3)\).
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Find the value of \(\int_{0}^{\pi/2} \cos^4\theta\,d\theta\). [5]
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Substitute the identity for \(\cos^4\theta\) from part (c) into the integral and integrate term-by-term.
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The integral evaluates to \(\frac{3\pi}{16}\).
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Describe the geometric transformation \(T\) defined by \(T(u)=wu\), where \(w = \cos\alpha + i\sin\alpha\). [2]
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Consider how multiplication by a complex number in polar form affects a point in the complex plane.
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The transformation \(T(u)=wu\) represents a rotation about the origin by an angle \(\alpha\).
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Find the complex numbers representing A’, B’, and C’ when A, B, C (given by \(1\), \(1+i\), and \(i\) respectively) are transformed by \(T\) with \(w = \cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})\). [4]
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Multiply each point by \(w\) to get its image.
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A’ = \(\frac{1}{2} + i\frac{\sqrt{3}}{2}\); B’ = \((\frac{1}{2} – \frac{\sqrt{3}}{2}) + i(\frac{1}{2} + \frac{\sqrt{3}}{2})\); C’ = \(-\frac{\sqrt{3}}{2} + i\frac{1}{2}\).
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Show that triangle A’B’C’ is congruent to triangle ABC. [5]
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Demonstrate that the distances between the corresponding points remain unchanged under the rotation.
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Since rotation preserves distances and angles, the side lengths of triangle A’B’C’ are equal to those of triangle ABC, proving the triangles are congruent.
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