# What is the polar form of ( -11,24 )?

Dec 2, 2015

Cartesian form: $\left(- 11 , 24\right) \iff$ polar form: $\left(\sqrt{597} , \pi + \arctan \left(- \frac{24}{11}\right)\right)$

#### Explanation: The radius is given by the Pythagorean Theorem as
$\textcolor{w h i t e}{\text{XXX}} r = \sqrt{{\left(- 11\right)}^{2} + {24}^{2}} = \sqrt{597}$

$\tan \left(\theta\right) = \frac{24}{- 11}$

Unfortunately, $\arctan$, the inverse of $\tan \left(\theta\right)$ is by definition an angle $\in \left(- \frac{\pi}{2} , + \frac{\pi}{2}\right)$ (i.e. in QI or Q IV)
[In the diagram above, the angle returned by $\arctan \left(- \frac{24}{11}\right)$ is shown as ${\theta}_{2}$.]

Taking $\arctan \left(\frac{24}{- 11}\right) = \arctan \left(\frac{- 24}{11}\right)$
gives us an angle that is "pi" radians less than the true value of $\theta$

Therefore $\theta = \pi + \arctan \left(- \frac{24}{11}\right)$

$\left(r , \theta\right) = \left(\sqrt{597} , \pi + \arctan \left(- \frac{24}{11}\right)\right)$

(using a calculator)
$\textcolor{w h i t e}{\text{XXX}} \approx \left(24.4 , 2.0\right) \approx \left(24.4 , {114.6}^{\circ}\right)$