Question

A 15-foot ladder is leaning against a tree. The base of the ladder is 8 feet away from the tree base. Using the inverse of sine, what is the approximate angle between the ladder and the tree? (1 point)
Responses

32°
32°

25°
25°

58°
58°

90°

Answer 1

To find the angle between the ladder and the tree, we can use the sine function. The ladder acts as the hypotenuse of a right triangle, the distance from the base of the ladder to the tree is one side (8 feet), and the height up the tree where the ladder touches is the other side.

Let's denote:

- \( L \) (the length of the ladder) = 15 feet (hypotenuse)
- \( d \) (the distance from the base of the ladder to the tree) = 8 feet (adjacent side)
- \( h \) (the height up the tree) = unknown (opposite side)

We want to find the angle \( \theta \) between the ladder and the ground. Using the definition of sine in a right triangle:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]

Alternatively, we can find the cosine of the angle, since we know the adjacent side:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{d}{L} = \frac{8}{15}
\]

Now we can use the inverse cosine to find the angle \( \theta \):

\[
\theta = \cos^{-1}\left(\frac{8}{15}\right)
\]

Calculating this:

\[
\theta \approx \cos^{-1}(0.5333) \approx 58°
\]

Thus, the approximate angle between the ladder and the tree is **58°**.