# How do you determine the exact coordinates of a point on the terminal arm of the angle in standard position given 45 degrees?

Feb 26, 2018

See below.

#### Explanation:

By looking at the diagram we can find a relationship between the Cartesian coordinates and the angular measurement.

Point $\boldsymbol{P}$ has Carteaian coordinates $\left(x , y\right)$, and polar coordinates $\left(r , \theta\right)$, we are only concerned with Cartesian for this. We can see that these correspond to the sine and cosine functions in the following way.

$x = r \cos \left(\theta\right)$

$y = r \sin \left(\theta\right)$

Where $\boldsymbol{r}$ is the radius, for a unit circle this will be $\boldsymbol{1}$. The cordinates of $\boldsymbol{P}$ are now:

$\left(r \cos \left(\theta\right) , r \sin \left(\theta\right)\right)$

So using this idea:

For:

radius $\boldsymbol{1}$ and $\theta = {45}^{\circ}$, we have:

$x = \cos \left({45}^{\circ}\right) = \frac{\sqrt{2}}{2}$

$y = \sin \left({45}^{\circ}\right) = \frac{\sqrt{2}}{2}$

So coordinates are:

$\left(\frac{\sqrt{2}}{2} , \frac{\sqrt{2}}{2}\right)$