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

Answer 1

To find the angle between the ladder and the tree, we can use the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.

In this scenario, we have:
- The length of the ladder (the hypotenuse) is 15 feet.
- The distance from the base of the ladder to the tree (the base of the triangle) is 8 feet.
- We need to determine the height of the point where the ladder touches the tree (the opposite side).

First, we can find the height using the Pythagorean theorem:

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

Now, we can find the angle \( \theta \) between the ladder (the hypotenuse) and the tree (the opposite side) using the sine function:

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

Now, calculate \( \theta \) using the inverse sine (arcsin):

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

Using a calculator to find the value:

\[
\theta \approx \arcsin\left(\frac{12.6886}{15}\right) \approx \arcsin(0.8459) \approx 57.59^\circ
\]

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

\[
\theta \approx 57.59^\circ
\]

Therefore, the response is approximately **57.6 degrees**.