It is given that $z=1+\sqrt{3}\,\mathbf{i}$ is a root of the equation $3{{z}^{3}}+a{{z}^{2}}+bz-8=0$, where $a$ and $b$ are real numbers. Find the exact values of $a$ and $b$ and hence solve the equation completely, giving all the roots in exact form. [5] Using Euler’s formula that ${{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $, show that ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{4}}=\cos 4\theta +\mathbf{i}\sin 4\theta $. [1] Using result shown in (i), by comparing the real parts, show that $\cos 4\theta ={{\cos }^{4}}\theta -6{{\cos }^{2}}\theta {{\sin }^{2}}\theta +{{\sin }^{4}}\theta $. Obtain an expression for $\sin 4\theta $ in terms of $\cos \theta $ and $\sin \theta $. Hence show that $\tan 4\theta =\frac{4\tan \theta -4{{\tan }^{3}}\theta }{1-6{{\tan }^{2}}\theta +{{\tan }^{4}}\theta }$. [4] Hence, find the possible values of $\tan \theta $ given that $\tan 4\theta =4$, leaving your answers correct to $3$ significant figures. [3]

2021 SAJC P1 Q7

13 marks

Prelims

It is given that $z=1+\sqrt{3}\,\mathbf{i}$ is a root of the equation $3{{z}^{3}}+a{{z}^{2}}+bz-8=0$, where $a$ and $b$ are real numbers. Find the exact values of $a$ and $b$ and hence solve the equation completely, giving all the roots in exact form.[5]
1. Using Euler’s formula that ${{\mathrm{e}}^{\mathbf{i}\theta }}=\cos \theta +\mathbf{i}\sin \theta $, show that ${{\left( \cos \theta +\mathbf{i}\sin \theta \right)}^{4}}=\cos 4\theta +\mathbf{i}\sin 4\theta $.[1]
2. Using result shown in (i), by comparing the real parts, show that $\cos 4\theta ={{\cos }^{4}}\theta -6{{\cos }^{2}}\theta {{\sin }^{2}}\theta +{{\sin }^{4}}\theta $.
  Obtain an expression for $\sin 4\theta $ in terms of $\cos \theta $ and $\sin \theta $. Hence show that $\tan 4\theta =\frac{4\tan \theta -4{{\tan }^{3}}\theta }{1-6{{\tan }^{2}}\theta +{{\tan }^{4}}\theta }$.[4]
3. Hence, find the possible values of $\tan \theta $ given that $\tan 4\theta =4$, leaving your answers correct to $3$ significant figures.[3]

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