How do you find the polar coordinates given the rectangular coordinates (-1/4, -sqrt3/4)?

1 Answer
Aug 23, 2017

See explanation...

Explanation:

You can plot -1/4, -sqrt(3/4) on the Cartesian plane easily enough. We seek a representation of this same point in (2d) space using the polar coordinates (r, theta)

The r is easy enough:

r = sqrt((-1/4)^2 + (-sqrt(3/4))^2

= sqrt(1/16 + 3/4)

= sqrt(13)/4

The value we calculate for the angle theta will depend on where we choose the theta = 0 direction to be. This is an arbitrary choice. Here I will choose theta = 0 to be on the x-axis to the right. So, therefore, theta = pi/2 will be on the y-axis pointing upwards, etc.

If r = sqrt(13)/4, then (-1/4)/(sqrt(13)/4) = cos(theta)

so therefore theta = arccos(-1/sqrt(13))

...but note that you can choose the theta = 0 direction to be on the y axis pointing up. In this case,

(-1/4)/(sqrt(13)/4) = sin(theta).

and therefore theta would be arcsin(-1/sqrt(13))

...I'm guessing your teacher may have a preferred direction for the theta = 0 direction, so use whichever of these answers is appropriate.

GOOD LUCK.