A ramp that is being constructed must be 42 inches long and reach a ledge that is 21 inches tall. Which inverse trigonometric ratio could be used to find [tex]m \angle B[/tex]?

[tex]\[
\begin{array}{l}
\sin^{-1}\left(\frac{21}{42}\right) \\
\sin^{-1}\left(\frac{42}{21}\right) \\
\cos^{-1}\left(\frac{21}{42}\right) \\
\sin^{-1}\left(\frac{21 \sqrt{3}}{42}\right)
\end{array}
\][/tex]



Answer :

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]