Question

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

Answer 1

The polar coordinates, in `(r, theta)` form, are `(sqrt8, -0.786)`. This is equivalent to `(sqrt8, (7pi)/4)`.

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((-2^2)+(-2^2)) = sqrt8`

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 = (opposite)/"hypotenuse" = -2/sqrt8`

Use your calculator, ensuring it is on radians rather than degrees mode, to find the angle whose sin is `-2/sqrt8: -0.786` `radians`.

This means that the polar coordinates, in `(r, theta)` form, are `(sqrt8, -0.786)`.

It's worth noting that this is equivalent to `(sqrt8, (7pi)/4)`.