# How do you write 2+3i in trigonometric form?

##### 1 Answer
Apr 9, 2017

To convert, $a + b i \to r \left(\cos \left(\theta\right) + i \sin \left(\theta\right)\right)$, use:
$r = \sqrt{{a}^{2} + {b}^{2}}$
$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right) + 0 , \pi , \pi , 2 \pi$
Depending on whether the signs of "a" and "b" indicate the, 1st, 2nd, 3rd or 4th quadrant.

#### Explanation:

Given: $2 + 3 i$

$r = \sqrt{{2}^{2} + {3}^{2}}$

$r = \sqrt{4 + 9}$

$r = \sqrt{13}$

The signs of "a" and "b" indicate the 1st quadrant:

$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right) + 0$

$\theta = {\tan}^{-} 1 \left(\frac{3}{2}\right)$

$\theta \approx {56.3}^{\circ} \mathmr{and} 0.983 \text{ radians}$

$2 + 3 i \to \sqrt{13} \left(\cos \left({56.3}^{\circ}\right) + i \sin \left({56.3}^{\circ}\right)\right)$