# What is the polar form of (-3,4)?

Jan 6, 2016

$\left(5 , {126.87}^{0}\right)$

#### Explanation:

If $\left(a , b\right)$ is a are the coordinates of a point in Cartesian Plane, $u$ is its magnitude and $\alpha$ is its angle then $\left(a , b\right)$ in Polar Form is written as $\left(u , \alpha\right)$.
Magnitude of a cartesian coordinates $\left(a , b\right)$ is given by$\sqrt{{a}^{2} + {b}^{2}}$ and its angle is given by ${\tan}^{-} 1 \left(\frac{b}{a}\right)$

Let $r$ be the magnitude of $\left(- 3 , 4\right)$ and $\theta$ be its angle.
Magnitude of $\left(- 3 , 4\right) = \sqrt{{\left(- 3\right)}^{2} + {4}^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5 = r$
Angle of $\left(- 3 , 4\right) = T a {n}^{-} 1 \left(\frac{4}{-} 3\right) = T a {n}^{-} 1 \left(- \frac{4}{3}\right) = - {53.13}^{0}$
$\implies$ Angle of $\left(- 3 , 4\right) = - {53.13}^{0}$

But the given point (-3,4) is in second quadrant so we have to add ${180}^{0}$ in the angle.
$\implies$ Angle of $\left(- 3 , 4\right) = - {53.13}^{0} + {180}^{0} = {126.87}^{0} = \theta$

$\implies \left(- 3 , 4\right) = \left(r , \theta\right) = \left(5 , {126.87}^{0}\right)$
$\implies \left(- 3 , 4\right) = \left(5 , {126.87}^{0}\right)$
Note that the angle is given in degree measure.