Question

Sarah is looking at the top of the street light. Her line of sight, which is the diagonal distance to the top of the street light, is 12 feet. She is standing 9 feet from the base of the light.

Use the inverse of cosine to find the angle of elevation with which she is looking at the light.

Answer 1

To find the angle of elevation \( \theta \) with which Sarah is looking at the top of the street light, we can use the cosine function from trigonometry.

Given:
- The diagonal distance (hypotenuse) from Sarah to the top of the street light is 12 feet.
- The horizontal distance from Sarah to the base of the street light is 9 feet.

The cosine of the angle \( \theta \) can be calculated using the formula:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Here, the adjacent side is the horizontal distance (9 feet), and the hypotenuse is the diagonal line of sight (12 feet). So we can write:

\[
\cos(\theta) = \frac{9}{12}
\]

Now, we simplify this:

\[
\cos(\theta) = \frac{3}{4}
\]

To find the angle \( \theta \), we use the inverse cosine function:

\[
\theta = \cos^{-1}\left(\frac{3}{4}\right)
\]

Using a calculator:

\[
\theta \approx 41.41^\circ
\]

Thus, the angle of elevation with which Sarah is looking at the light is approximately \( 41.41^\circ \).