So, we want to express the following:
[tex]\sec (\sin ^{-1}(\frac{x}{\sqrt[]{x^2+81}}))[/tex]
As an algebraic expression.
If:
[tex]\begin{gathered} \sin ^{-1}(\frac{x}{\sqrt[]{x^2+81}})=\theta \\ \text{Then,} \\ \sin (\theta)=\frac{x}{\sqrt[]{x^2+81}} \end{gathered}[/tex]
We could draw the following triangle:
Remember that the secant function relations the hypotenuse of the triangle and the adjacent side of the triangle. So first, we should find the adjacent side using the pythagorean theorem:
[tex]\begin{gathered} a^2=(\sqrt[]{x^2+81})^2-x^2 \\ a^2=x^2+81-x^2 \\ a^2=81\to a=9 \end{gathered}[/tex]
Therefore, the adjacent side is 81. And, the value of:
[tex]\sec (\sin ^{-1}(\frac{x}{\sqrt[]{x^2+81}}))[/tex]
Is:
[tex]\sec (\sin ^{-1}(\frac{x}{\sqrt[]{x^2+81}}))=\frac{\sqrt[]{x^2+81}}{9}[/tex]
