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