How to calculate angle within a circle

Question

Given this situation where the circle is cut by the rectangle. How do you calculate the angle α

Answer 1

If you have a right triangle, $\sin$ is geometrically defined as the quotient of the length of the opposite side of the angle by the length of the hypotenuse.

Here, it is not clear that the side of length $d=80$ is perpendicular to the side marked as $r=130$. If it is the case, you can easily deduce that the height of the triangle is equal to $d=80$.

Let $\vert XY\vert$ denote the length of a line segment determined by some points $X$ and $Y$.

If the height of the triangle $\triangle ABC$ is $CH$, we have:

$$\sin\alpha=\frac{\vert CH\vert}{\vert AC\vert}$$

by the geometric definition of $\sin$ since the triangle $\triangle AHC$ is a right triangle in $H$ and $AC$ is its hypotenuse.

How can you determine $\vert AC\vert$? Just observe that $AC$ is a radius of the circle, hence $\vert AC\vert = 130$.

Hence, you get:

$$\sin \alpha =\frac{\vert CH\vert}{\vert AC\vert}=\frac{80}{130}=\frac{8}{13}$$

As it is not a particular value, you need the help of a calculator.

$$\alpha = \sin^{-1}\left(\frac{8}{13}\right)\approx 0.662873824 \text{ rad}$$

which is about $38^{\circ}$.

However, be aware that the $\sin^{-1}$ functionality has to be used precautiously, but as JeanMarie put it in the comment section, it is too elaborate for your level.