# How do you convert 1 + 4i to polar form?

May 15, 2016

$\cong \sqrt{17} \left(\cos 1.33 + i \sin 1.33\right)$

#### Explanation:

If $z = a + b i$

Then $z$ may be expressed in polar form as:

$z = r \left(\cos \theta + i \sin \theta\right)$
Where: $r = \sqrt{{a}^{2} + {b}^{2}}$ and $\theta = \arctan \left(\frac{b}{a}\right)$

In this example: $z = 1 + 4 i$
Hence: $a = 1$ and $b = 4$

Therefore: $r = \sqrt{{1}^{2} + {4}^{2}}$ =$\sqrt{17}$
And: $\theta = \arctan \left(\frac{4}{1}\right)$ $\cong 1.33$

Hence: $z \cong \sqrt{17} \left(\cos 1.33 + i \sin 1.33\right)$