Let's solve the problem step-by-step:
1. Identify the given values:
- The length of the ramp (which can be considered the hypotenuse in a right triangle) is 42 inches.
- The height of the ledge (which can be considered the opposite side to the angle [tex]\( \angle B \)[/tex]) is 21 inches.
2. Determine which trigonometric function to use:
- Since we are given the opposite side and the hypotenuse, we should use the sine function. The sine function relates the angle to the opposite side and the hypotenuse in a right triangle:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
- Here, [tex]\(\theta = \angle B\)[/tex], the opposite side is 21 inches, and the hypotenuse is 42 inches.
3. Set up the equation for sine:
[tex]\[
\sin(\angle B) = \frac{21}{42}
\][/tex]
4. Simplify the fraction:
[tex]\[
\sin(\angle B) = \frac{21}{42} = \frac{1}{2}
\][/tex]
5. Finding the angle using the inverse sine function:
- To find [tex]\(\angle B\)[/tex], we use the inverse sine function [tex]\( \sin^{-1} \)[/tex]:
[tex]\[
\angle B = \sin^{-1}\left(\frac{1}{2}\right)
\][/tex]
6. Identify the correct inverse trigonometric ratio from the options given:
- We need [tex]\( \sin^{-1}\left(\frac{21}{42}\right) \)[/tex], which simplifies to [tex]\( \sin^{-1}\left(\frac{1}{2}\right) \)[/tex].
Hence, the correct choice is:
\[
\sin ^{-1}\left(\frac{21}{42}\right)
\}
