Question

In Problems $58-63$, find the exact value of each expression.

$\sin \left({\cos }^{-1}\frac{5}{13}-{\cos }^{-1}\frac{4}{5}\right)$.

Short answer

The exact value of the expression$\sin \left({\cos }^{-1}\frac{5}{13}-{\cos }^{-1}\frac{4}{5}\right)$is,$\frac{33}{65}$.

Step-by-step solution

Step 1 Given expression is,

$\sin \left({\cos }^{-1}\frac{5}{13}-{\cos }^{-1}\frac{4}{5}\right)$.

The formulae used in this problem are,

$$\begin{gathered} {\sin }^2\alpha +{\cos }^2\alpha =1 \\ \sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \end{gathered}$$

Let $\alpha ={\cos }^{-1}\frac{5}{13}$

$\cos \alpha =\frac{5}{13}$

Step 2 Find the value of sin α.

$$\begin{gathered} \sin \alpha =\sqrt{1-{\cos }^2\alpha } \\ =\sqrt{1-{\left(\frac{5}{13}\right)}^2} \\ =\sqrt{1-\frac{25}{169}} \\ =\sqrt{\frac{144}{169}} \\ \sin \alpha =\frac{12}{13} \end{gathered}$$

Step 3 Next find the value of cos β.

$$\begin{gathered} \cos \beta =\sqrt{1-{\sin }^2\alpha } \\ =\sqrt{1-{\left(\frac{4}{5}\right)}^2} \\ =\sqrt{1-\frac{16}{25}} \\ \cos \beta =\frac{3}{5} \end{gathered}$$

Step 4 Substitute the known values in the cos α-β formula.

$$\begin{gathered} \cos \left(\alpha -\beta \right)=\left(\frac{12}{13}\right)\left(\frac{4}{5}\right)-\left(\frac{5}{13}\right)\left(\frac{3}{5}\right) \\ =\frac{48}{65}-\frac{15}{65} \\ =\frac{33}{65} \\ \sin \left({\cos }^{-1}\frac{5}{13}-{\cos }^{-1}\frac{4}{5}\right)=\frac{33}{65} \end{gathered}$$