# What is the polar form of ( -6,36 )?

May 1, 2018

$\text{ }$
Cartesian Coordinates to Polar Form: color(blue)((-6, 36) = (36, 99.4^@)

#### Explanation:

$\text{ }$
Given the Cartesian Form: $\left(- 6 , 36\right)$

Find the Polar Form:color(blue)((r,theta)

color(green)("Step 1:"

Let us examine some of the relevant formula in context:

color(green)("Step 2:"

Plot the coordinate point color(blue)((-6, 36) on a Cartesian coordinate plane:

Indicate the known values, as appropriate:

$\overline{O A} = 6 \text{ Units}$

$\overline{A B} = 36 \text{ Units}$

Let $\overline{O B} = r \text{ Units}$

$\angle O A B = {90}^{\circ}$

Let $\angle A O B = {\alpha}^{\circ}$

color(green)("Step 3:"

Use the formula: color(red)(x^2 + y^2=r^2 to find color(blue)(r

Consider the following triangle with known values:

${r}^{2} = {6}^{2} + {36}^{2}$

$\Rightarrow 36 + 1296$

$\Rightarrow 1332$

${r}^{2} = 1332$

Hence, color(brown)(r=sqrt(1332)~~36.4966

To find the value of $\textcolor{red}{\theta}$:

$\tan \left(\theta\right) = \frac{36}{6} = 6$

$\theta = {\tan}^{-} 1 \left(6\right)$

$\theta \approx {80.53767779}^{\circ}$

color(blue)("Important:"

Since the angle color(red)(theta lines in Quadrant-II, we must subtract this angle from ${\textcolor{red}{180}}^{\circ}$ to get the required angle color(blue)(beta.

color(green)("Step 4:"

color(blue)(beta ~~ 180^@ - 80.53767779^@

$\Rightarrow \beta \approx {99.46232221}^{\circ}$

Hence, the required Polar Form:

color(blue)((r, theta) = (36, 99.4^@)

Hope it helps.