Inverse Sine

So far the angle was known and a side was not. Turn it round: a right triangle has an opposite side of 5 and a hypotenuse of 13, and you want the angle $\theta$. The ratio is easy to write, $\sin\left(\theta\right) = \frac{5}{13}$, but which angle has a sine of 5/13?

The inverse sine answers that question. $\arcsin$(5/13), also written $\sin^{-1}$(5/13), is the angle whose sine is 5/13: about 22.62°. The −1 is a name, not a power: sin⁻¹ x is an angle, never 1 divided by sin x.

On the board, after Sine ratio writes $\sin\left(\theta\right) = \frac{5}{13}$, the chip Use arcsin: every angle in the domain writes $\theta = \arcsin\left(\frac{5}{13}\right)$, and the round chip turns it into a number of degrees. The goal row says $\theta$ lies between 0° and 90°, because an acute angle of a right triangle always does.