# How do you convert the rectangular coordinate (-4.26,31.1) into polar coordinates?

Oct 10, 2016

$\left(31.3 , \frac{\pi}{2}\right)$

#### Explanation:

Changing to polar coordinates means we have to find $\textcolor{g r e e n}{\left(r , \theta\right)}$.

Knowing the relation between rectangular and polar coordinates that says:
$\textcolor{b l u e}{x = r \cos \theta \mathmr{and} y = r \sin \theta}$

Given the rectangular coordinates:
$x = - 4.26 \mathmr{and} y = 31.3$

${x}^{2} + {y}^{2} = {\left(- 4.26\right)}^{2} + {\left(31.3\right)}^{2}$
$\textcolor{b l u e}{{\left(r \cos \theta\right)}^{2}} + \textcolor{b l u e}{{\left(r \sin \theta\right)}^{2}} = 979.69$
${r}^{2} {\cos}^{2} \theta + {r}^{2} {\sin}^{2} \theta = 979.69$
${r}^{2} \left({\cos}^{2} \theta + {\sin}^{2} \theta\right) = 979.69$
Knowing the trigonometric identity that says:
$\textcolor{red}{{\cos}^{2} \theta + {\sin}^{2} \theta = 1}$
We have:

${r}^{2} \cdot \textcolor{red}{1} = 979.69$
$r = \sqrt{979.69}$
$\textcolor{g r e e n}{r = 31.3}$

Given:
$\textcolor{b l u e}{y} = 31.3$
$\textcolor{b l u e}{r \sin \theta} = 31.3$
$\textcolor{g r e e n}{31.3} \cdot \sin \theta 31.3$
$\sin \theta = \frac{31.3}{31.3}$
$\sin \theta = 1$
$\textcolor{g r e e n}{\theta = \frac{\pi}{2}}$

Therefore,the polar coordinates are
$\left(\textcolor{g r e e n}{31.3 , \frac{\pi}{2}}\right)$