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°

90°
90°

58°
58°

25°

Answer 1

To find the angle between the ladder and the tree, we can use the sine function. In this scenario, we have a right triangle where:

- The length of the ladder is the hypotenuse (15 feet),
- The distance from the base of the ladder to the tree is one of the legs (8 feet), and
- The height of the ladder against the tree is the other leg.

To find the angle (let's call it θ), we can use the sine function:

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

In this case, the opposite side is the height (which we will need to determine) and the hypotenuse is 15 feet.

We can find the height of the ladder against the tree using the Pythagorean theorem:

\[
a^2 + b^2 = c^2
\]
where
- \(a\) is the distance from the base of the ladder to the tree (8 feet),
- \(b\) is the height of the ladder against the tree (unknown),
- \(c\) is the length of the ladder (15 feet).

Rearranging, we can find \(b\):

\[
b^2 = c^2 - a^2
\]

Calculating \(b\):

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

Now we can find the sine of the angle θ:

\[
\sin(θ) = \frac{b}{c} = \frac{12.6886}{15}
\]

Calculating the sine:

\[
\sin(θ) \approx 0.8459
\]

Now we can use the inverse sine (arcsin) to find θ:

\[
θ \approx \arcsin(0.8459)
\]
Using a calculator:

\[
θ \approx 58°
\]

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