Question

How do I use a right triangle to write the following expression as an algebraic expression?

Answer 1

So, we want to express the following:

`\sec (\sin ^(-1)(\frac{x}{\sqrt[]{x^2+81}}))`

As an algebraic expression.

If:

`\begin{gathered} \sin ^(-1)(\frac{x}{\sqrt[]{x^2+81}})=\theta \\ \text{Then,} \\ \sin (\theta)=\frac{x}{\sqrt[]{x^2+81}} \end{gathered}`

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:

`\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}`

Therefore, the adjacent side is 81. And, the value of:

`\sec (\sin ^(-1)(\frac{x}{\sqrt[]{x^2+81}}))`

Is:

`\sec (\sin ^(-1)(\frac{x}{\sqrt[]{x^2+81}}))=\frac{\sqrt[]{x^2+81}}{9}`