Question

How do you find the rectangular coordinate of `(sqrt 1, 225^o)`?

Answer 1

I will assume the original coordinates are polar. If `(r, theta)=(sqrt1, 225^o)` then `(x, y)= (-0.7, -0.7)`.

Explanation:

We use trigonometry to convert polar coordinates to rectangular. See this answer for a more detailed discussion:

http://socratic.org/questions/how-do-you-convert-the-polar-coordinate-4-5541-1-2352-into-cartesian-coordinates

In this case, the hypotenuse is the distance of the point from the origin, in this case `sqrt1` units, but the square root of 1 is just 1. The angle from the positive `x` axis is `225^o`.

`x= r cos theta = 1*cos 225 = -0.707`

`y= r sin theta = 1*sin 225 = -0.707`

That means the rectangular (sometimes called Cartesian) coordinates of the point in question are `(-0.707, -0.707)`, but we really shouldn't give answers more precise than the data we were given, so round the answer to `(-0.7, -0.7)`.

This makes sense, because `225^o` places the point in the 'third quadrant', where both its `x` and `y` values are negative numbers.