Question

use a right triangle to write the following expression as an algebraic expression. assume that x is positive and in the domain of the given inverse trigonometric function. tan(cos^(-1)9x). which of the following triangles is correct to write the given expression as an algebraic expression? a. triangle image with hypotenuse 9x and an angle theta b. triangle image with adjacent side 9x and an angle theta c. triangle image with adjacent side 9x and an angle theta d. triangle image with opposite side 1 and hypotenuse 9x and an angle theta

Explanation:

Step1: Recall inverse - cosine definition

Let $\theta=\cos^{-1}(9x)$. By the definition of the inverse - cosine function, $\cos\theta = 9x=\frac{9x}{1}$, where $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$ in a right - triangle.

Step2: Identify the sides of the right - triangle

In a right - triangle with an angle $\theta$, if $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} = 9x=\frac{9x}{1}$, the adjacent side to the angle $\theta$ is $9x$ and the hypotenuse is $1$.

The correct right - triangle to represent $\cos^{-1}(9x)$ has the adjacent side to the angle $\theta$ equal to $9x$ and the hypotenuse equal to $1$. Looking at the options, the triangle in option C has the adjacent side to the angle $\theta$ as $9x$ and the hypotenuse as $1$.

Answer:

C.