Question

Find the exact value of each expression.

$\sin \left({\sin }^{-1}\frac{3}{5}-{\cos }^{-1}\left(-\frac{4}{5}\right)\right)$

Short answer

The value of the expression is$-\frac{24}{25}$

Step-by-step solution

Step 1. Given information

The given expression is$\sin \left({\sin }^{-1}\frac{3}{5}-{\cos }^{-1}\left(-\frac{4}{5}\right)\right)$

Step 2. Solve the expression

Let $\alpha ={\sin }^{-1}\frac{3}{5}$and $\beta ={\cos }^{-1}\left(-\frac{4}{5}\right)$

So,

$\sin \alpha =\frac{3}{5},-\frac{\mathrm{\pi }}{2}\le \alpha \le \frac{\mathrm{\pi }}{2}$and $\cos \beta =-\frac{4}{5},0\le \beta \le \mathrm{\pi }$

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

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

Now,

$$\begin{gathered} \sin \left(\alpha -\beta \right) \\ =\sin \alpha ·\cos \beta -\cos \alpha ·\sin \beta \\ =\frac{3}{5}·\left(-\frac{4}{5}\right)-\frac{4}{5}·\frac{3}{5} \\ =-\frac{24}{25} \end{gathered}$$