[Math] Geometry Question: Calculating the length of the side of a triangle inside a circle.

circlesgeometry

Given this picture:

enter image description here

The radius of the circle is $30$ inches. The angle between $A$ and $B$ is $22.5^o$.

How would I calculate the distance (not along the arc, but straight line) from $B$ to $A$, as depicted by the red line?

Best Answer

Bisect the angle to get 2 right triangles with known hypotenuse and angles, then use $\sin$ to get the sides opposite the $22.5/2$ degree angles.

Or, use the triangle that's already there, having 2 angles equal to $(180-22.5)/2$ degrees, and apply the law of sines.

Or, apply the law of cosines immediately.

In case you want exact forms for the sines or cosines involved, you can use half/double angle formulas and the fact that $22.5=45/2$.

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