Question

Write each trigonometric expression as an algebraic expression containing $u$and $v$. Give the restrictions required on $u$and $v$.

role="math" localid="1646564067411" $\cos \left({\cos }^{-1}u+{\sin }^{-1}v\right)$

Short answer

We can write$\cos \left({\cos }^{-1}u+{\sin }^{-1}v\right)$as$u\sqrt{1-v^2}-v\sqrt{1-u^2}$, where$-1\le u\le 1,-1\le v\le 1$

Step-by-step solution

Step 1. Given information

The given expression is$\cos \left({\cos }^{-1}u+{\sin }^{-1}v\right)$

Step 2. Solve the expression

Let $\alpha ={\cos }^{-1}u$and $\beta ={\sin }^{-1}v$

So,

$\cos \alpha =u$where $0\le \alpha \le \mathrm{\pi }$and $-1\le u\le 1$

$\sin \beta =v$where $-\frac{\mathrm{\pi }}{2}\le \beta \le \frac{\mathrm{\pi }}{2}$and $-1\le v\le 1$

role="math" localid="1646564490741" $\cos \beta =\sqrt{1-v^2}$

$\sin \alpha =\sqrt{1-u^2}$

Now,

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