The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number

Question

The complex number $w$ has modulus $\sqrt2$ and argument $-\frac{\pi}{4}$. The complex number $z$ has modulus $2$ and argument $-\frac{\pi}{3}$. Give all arguments as principal arguments in radians. Find the modulus and principal argument of $wz$.[2] Express $w$ and $z$ in Cartesian form. Hence find the exact real and imaginary parts of $wz$.[4] Deduce that $$\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}.$$[2]

Accepted Answer

(a) $|wz|=2\sqrt2,\ \arg(wz)=-\frac{7\pi}{12}$ (b) $w=1-i,\ z=1-\sqrt3\,i,\ \mathrm{Re}(wz)=1-\sqrt3,\ \mathrm{Im}(wz)=-1-\sqrt3$ (c) $\cos\frac{7\pi}{12}=\frac{\sqrt2-\sqrt6}{4}$

A product giving an exact cosine value

A product giving an exact cosine value

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