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

58°
58°

32°
32°

90°
90°

25°
25°

hhgf

Answer 1

To find the angle between the ladder and the tree using the dimensions provided, you can use the inverse sine function (arcsin).

1. The length of the ladder (the hypotenuse) is 15 feet.
2. The distance from the base of the ladder to the tree (the adjacent side) is 8 feet.
3. To find the angle (\(θ\)) between the ladder and the ground (the base), you can use the sine function, which is defined as:

\[
\sin(θ) = \frac{\text{opposite}}{\text{hypotenuse}}
\]

4. To find the height at which the ladder touches the tree (the opposite side), you can use the Pythagorean theorem:

\[
\text{height}^2 + 8^2 = 15^2
\]

Calculating this gives:

\[
\text{height}^2 + 64 = 225
\]
\[
\text{height}^2 = 161
\]
\[
\text{height} = \sqrt{161} \approx 12.6886 \text{ feet}
\]

5. Now, we can find the sine of the angle \(θ\):

\[
\sin(θ) = \frac{\text{height}}{\text{hypotenuse}} = \frac{\sqrt{161}}{15}
\]

6. Now, use the inverse sine to find the angle:

\[
θ \approx \arcsin\left(\frac{\sqrt{161}}{15}\right)
\]

However, if we need to find the angle between the ladder and the tree, we actually need to work with the right triangle formed by the height and the base (the distance from the tree):

Using the cosine function:
- Base (adjacent) = 8 feet
- Hypotenuse = 15 feet

So the cosine of the angle is:

\[
\cos(θ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{15}
\]

Then taking the inverse cosine:

\[
θ = \arccos\left(\frac{8}{15}\right)
\]

Calculating this gives:

\[
θ \approx 58°
\]

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