To solve the problem of determining which inverse trigonometric ratio can be used to find [tex]\( m \angle B \)[/tex] for a ramp that is 42 inches long and reaches a ledge that is 21 inches tall, we need to follow these steps:
1. Understand the Geometry:
- The ramp forms a right triangle where:
- The hypotenuse is the ramp length, which is 42 inches.
- The opposite side to [tex]\( \angle B \)[/tex] is the ledge height, which is 21 inches.
2. Identify the Appropriate Trigonometric Function:
- The sine function is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle. Therefore:
[tex]\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\text{ledge height}}{\text{ramp length}} = \frac{21}{42}
\][/tex]
3. Calculate the Sine:
- Substituting the values:
[tex]\[
\sin(\theta) = \frac{21}{42} = \frac{1}{2}
\][/tex]
4. Inverse Trigonometric Function:
- The inverse sine function ([tex]\(\sin^{-1}\)[/tex]) will give us the angle whose sine is [tex]\(\frac{1}{2}\)[/tex]. Hence, to find [tex]\(m \angle B\)[/tex], we use:
[tex]\[
\theta = \sin^{-1}\left(\frac{21}{42}\right)
\][/tex]
Therefore, the correct inverse trigonometric ratio to find [tex]\( m \angle B \)[/tex] is:
[tex]\[
\sin^{-1}\left(\frac{21}{42}\right)
\][/tex]
Thus, the correct choice from the given options is:
[tex]\[
\sin^{-1}\left(\frac{21}{42}\right)
\][/tex]
