Evaluating the Composition of a Sine with an Inverse Cosine

Beginning with the inside, we can say there is some angle such that $\theta ={\cos }^{-1}\left(\frac{4}{5}\right),$ which means $\cos \theta =\frac{4}{5},$ and we are looking for $\sin \theta .$ We can use the Pythagorean identity to do this.

Since $\theta ={\cos }^{-1}\left(\frac{4}{5}\right)$ is in quadrant I, $\sin \theta$ must be positive, so the solution is $\frac{3}{5}.$ See Figure 11.

Figure 11 Right triangle illustrating that if $\cos \theta =\frac{4}{5},$ then $\sin \theta =\frac{3}{5}$

We know that the inverse cosine always gives an angle on the interval $\left[0,\pi \right],$ so we know that the sine of that angle must be positive; therefore $\sin \left({\cos }^{-1}\left(\frac{4}{5}\right)\right)=\sin \theta =\frac{3}{5}.$