# What is the polar form of ( -18,-61 )?

Jan 14, 2018

=> color(red)( (sqrt(4045) , tan^(-1) (61/18) + pi ) " In radians"

=> color(red)( (sqrt(4045) , tan^(-1) (61/18) + 180^circ) " In degrees"

#### Explanation:

We must first use our knowledge of polar coordinates...

We see from this diagram, that:

color(red)(x = rcostheta

color(red)(y = rsintheta

color(red)( r^2 = r^2cos^2x + r^2 sin^2x = x^2 + y^2 " Using pythagerous"

$\implies \left(x , y\right) \equiv \left(r \cos \theta , r \sin \theta\right)$

So we can find $r$:

${r}^{2} = {\left(- 18\right)}^{2} + {\left(- 61\right)}^{2}$

$\implies {r}^{2} = 324 + 3721 = 4045$

$\implies r = \sqrt{4045}$

We can now find $\alpha$

$\implies \tan \alpha = \frac{61}{18}$

$\implies \alpha = {\tan}^{- 1} \left(\frac{61}{18}\right)$

Now we need the angle form the positive $x$ axis

So hence

$\theta = \alpha + \pi \text{ Radians}$

$\theta = \alpha + {180}^{\circ} \text{ Degrees}$

$\implies \theta = {\tan}^{- 1} \left(\frac{61}{18}\right) + \pi$

or $\implies \theta = {\tan}^{- 1} \left(\frac{61}{18}\right) + {180}^{\circ}$

=> color(red)( (sqrt(4045) , tan^(-1) (61/18) + pi ) " In radians"

=> color(red)( (sqrt(4045) , tan^(-1) (61/18) + 180^circ) " In degrees"