Question

Express the complex number in trigonometric form. 3 - 3i

Answer 1

z=x+ iy
\(z=r ( \cos \theta + i \sin \theta ) \)
\(r =\sqrt {x^2+y^2}\)
\(x= r \cos \theta \)
\(y= r \cos \theta \)

Answer 2

i made a typo
\(y=r \sin \theta\)

Answer 3

\[r=\sqrt{3^2+(-3)^2}=\sqrt{18}=3\sqrt2\]
\[3=3\sqrt2\cos\theta~~\Rightarrow~~\cos\theta=\frac{1}{\sqrt2}~~\Rightarrow~~\theta=\frac{\pi}{4},\text{ or }\frac{7\pi}{4}\]
\[-3=3\sqrt2\sin\theta~~\Rightarrow~~\sin\theta=-\frac{1}{\sqrt2}~~\Rightarrow~~\theta=\frac{5\pi}{4},\text{ or }\frac{7\pi}{4}\]
Clearly the angle must be \(\dfrac{7\pi}{4}\), so
\[3-3i=3\sqrt2\left(\cos\frac{7\pi}{4}+i\sin\frac{7\pi}{4}\right)\]