Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number given $r = 13$ and $\tan θ = \frac{5}{12} .$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
If $\tan θ = \frac{5}{12},$ and $\tan θ = \frac{y}{x},$ we first determine $r = \sqrt{x^{2} + y^{2}} = \sqrt{12^{2} + 5^{2}} = 13 .$ We then find $\cos θ = \frac{x}{r}$ and $\sin θ = \frac{y}{r} .$
$\begin{array}{l} z = 13 (\cos θ + i \sin θ) \\ = 13 (\frac{12}{13} + \frac{5}{13} i) \\ = 12 + 5 i \end{array}$
The rectangular form of the given number in complex form is $12 + 5 i .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Convert the complex number to rectangular form:

z = 4 (\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})

How did it go?