# What is the trigonometric form of  (5+i) ?

Jun 12, 2016

$\sqrt{26} \left(\cos \left(0.197\right) + i \sin \left(0.197\right)\right)$

#### Explanation:

To convert a complex number into trig form.

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{x + y i = r \left(\cos \theta + i \sin \theta\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where r is the magnitude and $\theta$ the argument.

Since( 5 + i) is in the 1st quadrant the the following formulae allow us to calculate r and $\theta$

•r=sqrt(x^2+y^2

•theta=tan^-1(y/x)

here x = 5 and y = 1

$\Rightarrow r = \sqrt{{5}^{2} + {1}^{2}} = \sqrt{26}$

and $\theta = {\tan}^{-} 1 \left(\frac{1}{5}\right) = 0.197 \text{ radians} \mathmr{and} {11.3}^{\circ}$

$\Rightarrow 5 + i = \sqrt{26} \left(\cos \left(0.197\right) + i \sin \left(0.197\right)\right)$