Question

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

Answer 1

See below.

Explanation:

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

Point `bbP` has Carteaian coordinates `(x,y)`, and polar coordinates `(r,theta)`, 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=rcos(theta)`

`y=rsin(theta)`

Where `bbr` is the radius, for a unit circle this will be `bb1`. The cordinates of `bbP` are now:

`(rcos(theta),rsin(theta))`

So using this idea:

For:

radius `bb1` and `theta=45^@`, we have:

`x=cos(45^@)=sqrt(2)/2`

`y= sin(45^@)=sqrt(2)/2`

So coordinates are:

`(sqrt(2)/2,sqrt(2)/2)`