# How do you convert the Cartesian coordinates (-7,6) to polar coordinates?

##### 1 Answer
Oct 22, 2015

$\left(x , y\right) = \left(- 7 , 6\right) \iff \left(r , \theta\right) \cong \left(9.22 , 2.43\right)$

#### Explanation:

$\left(- 7 , 6\right)$ is a point in Quadrant II
and if $\theta$ is the angle between the positive X-axis and $\left(- 7 , 6\right)$ with vertex at the origin:
$\textcolor{w h i t e}{\text{XXX}} \tan \left(\theta\right) = - \frac{6}{7}$

Using a claculator
$\textcolor{w h i t e}{\text{XXX}} \hat{\theta} = \arctan \left(- \frac{6}{7}\right)$
gives us
$\textcolor{w h i t e}{\text{XXX}} \hat{\theta} = - 0.70863$

$\hat{\theta}$ is the same reference angle as the required $\theta$ but since $\arctan \left(_\right)$ is restricted to a range of $\left(- \frac{\pi}{2} , \frac{\pi}{2}\right]$
$\hat{\theta}$ is in Quadrant IV

Therefore
$\textcolor{w h i t e}{\text{XXX}} \theta = \pi + \arctan \left(- \frac{6}{7}\right) \cong 2.43$

$r$ (the radius) can be calculated using the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} r = \sqrt{{\left(- 7\right)}^{2} + {6}^{2}} \cong 9.22$