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

Jan 29, 2016

I will assume the original coordinates are polar. If $\left(r , \theta\right) = \left(\sqrt{1} , {225}^{o}\right)$ then $\left(x , y\right) = \left(- 0.7 , - 0.7\right)$.

#### 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 $\sqrt{1}$ units, but the square root of 1 is just 1. The angle from the positive $x$ axis is ${225}^{o}$.

$x = r \cos \theta = 1 \cdot \cos 225 = - 0.707$

$y = r \sin \theta = 1 \cdot \sin 225 = - 0.707$

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

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