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?

To find the angle between the ladder and the tree using the inverse of sine, we can use the following relationship based on the right triangle formed by the ladder, the ground, and the tree:

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

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 determine the height of the ladder against the tree (the opposite side). We can do this using the Pythagorean theorem:

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

Now we can find the angle \(\theta\):

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

Now calculate \(\sin(\theta)\):

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

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

\[ \theta = \arcsin(0.8459) \]

Using a calculator:

\[ \theta \approx 57.6^\circ \]

So the closest approximation from the list given is:

• 58.0°