2020HSCShort answer

Q13 (d) (ii)

3 marks

Question

By expanding $(e^{i\theta}+e^{-i\theta})^4$ , show that
$\cos^4\theta=\frac18\bigl(\cos(4\theta)+4\cos(2\theta)+3\bigr).$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Correctly uses binomial theorem and groups conjugate pairs, or equivalent merit	2
Uses part (a) to obtain $(e^{i\theta}+e^{-i\theta})^4=16\cos^4\theta$ <br>OR<br>Attempts to use binomial theorem, or equivalent merit	1

Sample answer:

(e^{i\theta}+e^{-i\theta})^4=e^{4i\theta}+4e^{3i\theta}\cdot e^{-i\theta}+6e^{2i\theta}\cdot e^{-2i\theta}+4e^{i\theta}\cdot e^{-3i\theta}+e^{-4i\theta}

=(e^{4i\theta}+e^{-4i\theta})+4(e^{2i\theta}+e^{-2i\theta})+6

=2\cos4\theta+8\cos2\theta+6

Also,

(e^{i\theta}+e^{-i\theta})^4=2^4\cos^4\theta

\therefore\cos^4\theta=\frac18(\cos4\theta+4\cos2\theta+3)

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Syllabus alignment

Working MathematicallyComplex Numbers

Other representations of complex numbersUse Euler's formula and convert between Cartesian, polar and exponential forms to solve problems.

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