2023TrialShort answer

Q13 (a) (ii)

2 marks

Question

By considering the roots of $z^3=4\sqrt{2}(1+i)$ , show that

\cos\frac{7\pi}{12}+\cos\frac{\pi}{12}=\cos\frac{\pi}{4}.

Try the question before revealing the answer.

Solution

\begin{aligned} \sum \bar{\alpha}&=2\left(\cos\frac\pi{12}+i\sin\frac\pi{12}\right)+2\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)\\ &\quad+2\left(\cos\left(-\frac{7\pi}{12}\right)+i\sin\left(-\frac{7\pi}{12}\right)\right)=0\\ \text{Equating real parts: }&\quad 2\cos\frac\pi{12}+2\cos\frac{3\pi}{4}+2\cos\left(-\frac{7\pi}{12}\right)=0\\ \cos\frac\pi{12}-\cos\frac\pi4+\cos\frac{7\pi}{12}&=0\\ \therefore\quad \cos\frac\pi{12}+\cos\frac{7\pi}{12}&=\cos\frac\pi4 \end{aligned}

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Complex Numbers

Solving equations with complex numbersUse De Moivre's theorem and solve equations and polynomial problems involving complex and conjugate roots.
Other representations of complex numbersUse Euler's formula and convert between Cartesian, polar and exponential forms to solve problems.

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