Accedmychevron_right11thchevron_rightmathchevron_rightComplex Numberschevron_rightExercise 1.5

(viii) (52,5π12)(-\dfrac{5}{2},\dfrac{5\pi}{12})

Here r=52<0r=-\dfrac{5}{2}<0 and θ=5π12\theta=\dfrac{5\pi}{12}.

Since r<0r<0, use angle:

5π12+π=17π12\frac{5\pi}{12}+\pi=\frac{17\pi}{12}

Use:

cos5π12=624,sin5π12=6+24\cos\frac{5\pi}{12}=\frac{\sqrt{6}-\sqrt{2}}{4},\quad \sin\frac{5\pi}{12}=\frac{\sqrt{6}+\sqrt{2}}{4} x=52cos5π12=5(62)8y=52sin5π12=5(6+2)8\begin{aligned} x&=-\frac{5}{2}\cos\frac{5\pi}{12}=-\frac{5(\sqrt{6}-\sqrt{2})}{8}\\ y&=-\frac{5}{2}\sin\frac{5\pi}{12}=-\frac{5(\sqrt{6}+\sqrt{2})}{8} \end{aligned}