# What are the coordinates for the point p(45^circ) where p(theta)=(x,y) is the point where the terminal arm of an angle theta intersects the unit circl?

Aug 3, 2018

$\left(\frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}}\right)$. See graphical depiction.

#### Explanation:

The polar equation of this unit circle is $r = 1$.

A point of the radial line $\vec{O P}$, in the direction $\theta$ are

$p \left(\theta\right) = r \left(\cos \theta , \sin \theta\right)$.

If P is the point of intersection with r = 1,

$p \left(\theta\right) = \left(\cos \theta , \sin \theta\right)$.. So,

$p \left(\frac{\pi}{4}\right) = \left(\cos \left(\frac{\pi}{4}\right) , \sin \left(\frac{\pi}{4}\right)\right) = \left(\frac{1}{\sqrt{2}} , \frac{1}{\sqrt{2}}\right)$.

See graphical depiction.

graph{(x^2+y^2-1)(y-x)((x-1/sqrt2)^2+(y-1/sqrt2)^2-0.001)=0[0 2 0 1]}