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. cos(sin^(-1)(5/x)). which of the following triangles is correct to write the given expression as an algebraic expression? a. description of triangle with angle θ, side 5, hypotenuse x b. description of triangle with angle θ, side 5, hypotenuse x c. description of triangle with angle θ, side x, hypotenuse 5 d. description of triangle with angle θ, side 5, hypotenuse x

Explanation:

Step1: Recall the definition of inverse - sine function

If $\theta=\sin^{-1}\frac{5}{x}$, then $\sin\theta=\frac{5}{x}$. In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So, the side opposite to the angle $\theta$ has length $5$ and the hypotenuse has length $x$.

Step2: Analyze the right - triangle options

In a right - triangle with an angle $\theta$ such that $\sin\theta = \frac{5}{x}$, the side opposite to $\theta$ is $5$ and the hypotenuse is $x$. Looking at the options, in option B, the side opposite to the angle $\theta$ is $5$ and the hypotenuse is $x$.

Answer:

B.