# How do I find the trigonometric form of the complex number 3-4i?

Mar 22, 2018

The trigonometric form is $= 5 \left(\cos \left(- 0.93\right) + i \sin \left(- 0.93\right)\right)$

#### Explanation:

Any complex number

$z = a + i b$

can be converted to the polar form

$z = | z | \left(\cos \theta + i \sin \theta\right)$

Where,

$\cos \theta = \frac{a}{| z |}$

and

$\sin \theta = \frac{b}{| z |}$

Here,

$z = 3 - 4 i$

$| z | = \sqrt{{\left(3\right)}^{2} + {\left(- 4\right)}^{2}} = \sqrt{25} = 5$

$\cos \theta = \frac{3}{5} = 0.6$

$\sin \theta = - \frac{4}{5} = - 0.8$

Therefore,

$\theta = \arcsin \left(- 0.8\right) = - 0.93 r a d$, $\left[\mod 2 \pi\right]$

$z = 5 \left(\cos \left(- 0.93\right) + i \sin \left(- 0.93\right)\right)$