# APPENDIX H - Complex Numbers - H Exercises: 25

$3\displaystyle \sqrt{2}(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4})$

#### Work Step by Step

We are given: $z=-3+3i$ To find $r$ of a complex number $a+bi$, we use: $\sqrt{a^2+b^2}$: $r=\sqrt{(-3)^{2}+3^{2}}=3\sqrt{2}$ To find $\theta$, we use $\tan{\theta}=\frac{b}{a}$: $\displaystyle \tan\theta=\frac{3}{-3}=-1$ And since $z$ lies in the second quadrant, we have: $\displaystyle \theta=\frac{3\pi}{4}$ To put the number in polar form, we use $r(\cos{\theta}+i\sin{\theta})$: $3\displaystyle \sqrt{2}(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4})$

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