# How do you convert (-2,-2) into polar coordinates?

Jan 29, 2016

The polar coordinates, in $\left(r , \theta\right)$ form, are $\left(\sqrt{8} , - 0.786\right)$. This is equivalent to $\left(\sqrt{8} , \frac{7 \pi}{4}\right)$.

#### Explanation:

When converting from polar to rectangular coordinates, we can use:

$x = r \cos \theta$

$y = r \sin \theta$

Going in the opposite direction, our first step is to find $r$, the radius of the circle:

$r = \sqrt{{x}^{2} + {y}^{2}} = \sqrt{\left(- {2}^{2}\right) + \left(- {2}^{2}\right)} = \sqrt{8}$

Now we know the radius, and this is the hypotenuse of a right-angled triangle with the other two sides being $x = - 2$ (adjacent) and $y = - 2$ (opposite). We can use the definition of trig functions to find the value of $\theta$:

$\sin \theta = \frac{o p p o s i t e}{\text{hypotenuse}} = - \frac{2}{\sqrt{8}}$

Use your calculator, ensuring it is on radians rather than degrees mode, to find the angle whose sin is $- \frac{2}{\sqrt{8}} : - 0.786$ $r a \mathrm{di} a n s$.

This means that the polar coordinates, in $\left(r , \theta\right)$ form, are $\left(\sqrt{8} , - 0.786\right)$.

It's worth noting that this is equivalent to $\left(\sqrt{8} , \frac{7 \pi}{4}\right)$.