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

25°
25°

32°
32°

58°
58°

90°

Answer 1

To find the angle between the ladder and the tree using the inverse sine function, we can use the sine definition in a right triangle. The ladder serves as the hypotenuse, and the height of the ladder above the ground (the distance between the top of the ladder and the ground, which is the tree) and the distance from the base of the ladder to the tree form the two legs of the triangle.

We know:
- Hypotenuse (ladder) = 15 feet
- Opposite side (height of ladder against tree) = using the Pythagorean theorem we can calculate this.
- Adjacent side (base distance from tree) = 8 feet

Using the Pythagorean theorem, we find the height (opposite side):

\[
h^2 + 8^2 = 15^2
\]
\[
h^2 + 64 = 225
\]
\[
h^2 = 225 - 64
\]
\[
h^2 = 161
\]
\[
h = \sqrt{161} \approx 12.6886 \text{ feet}
\]

Now we can find the angle \( \theta \) using the inverse sine:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{15}
\]
\[
\sin(\theta) = \frac{\sqrt{161}}{15}
\]

Calculating this value:

\[
\sin(\theta) \approx \frac{12.6886}{15} \approx 0.8459
\]

Now calculate the angle \( \theta \):

\[
\theta \approx \sin^{-1}(0.8459)
\]

Using a calculator:

\[
\theta \approx 58°
\]

Therefore, the approximate angle between the ladder and the tree is:

**58°**.