# Distance from a point on a circle to another when the angle between them is given

There are two points on a circle and the angle between them is known, as well as the radius of the circle. What I want to do is find the horizontal and vertical distance between these points. Is there any way to do that?(the distance given by the horizontal&vertical yellow lines)

Using the law of sines you easily get $$d= \sqrt 2 r \sqrt {1 - \cos x}$$

where d is the hypotenuse of your yellow triangle.

If the radius of the circle is r, then using the law of cosines the length of the chord joining the two points is

\begin{align*} c &= \sqrt{r^2 + r^2 - 2r^2\cos{x}} \\ &= r\sqrt{2(1 - \cos{x})} \end{align*}

However, to find the horizontal and vertical components of the distance you need to know the location of the points. This is because if you imagine the same angle in a different position on the circle (such as A and B below), the chord length is the same but the horizontal and vertical components are different. Here is a diagram made with Isosceles:

Better to use $t$ or $\theta$ for the angle so as not to confuse with $x$ coordinate.

Top Start Point

$$(x,y)= (0,r)$$

End Point

$$(x,y)= r [ \sin t, \cos t]$$

Horizontal and Vertical distances of your yellow lines ( please thicken them ), are $x$ and $y$ distance differences respectively.

Use the distance relation

$$d^2 = 2 \, r^2 (1- \cos t), \quad d = r\sqrt{2 (1- \cos t) }.$$