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

90°
90°

32°
32°

25°
25°

58°

Answer 1

To find the angle θ between the ladder and the tree using the inverse of sine (arcsin), we can use the following relationship based on a right triangle:

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

In this case:
- The opposite side is the height of the ladder against the tree, which we can find using the Pythagorean theorem.
- The hypotenuse is the length of the ladder (15 feet).
- The base (adjacent side) is 8 feet.

First, we need to calculate the height (opposite side) using the Pythagorean theorem:

\[
a^2 + b^2 = c^2
\]
Where:
- \( a \) is the height against the tree (opposite side),
- \( b \) is the base (8 feet),
- \( c \) is the length of the ladder (15 feet).

Rearranging this gives us:

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

Now, we can find the sine of the angle θ:

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

Now we find the angle using inverse sine:

\[
\theta = \arcsin(0.8459) \approx 58°.
\]

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