To find the angle between the ladder and the tree using the inverse sine function, we can use the relationship in a right triangle formed by the ladder, the ground, and the tree.
- The length of the ladder (hypotenuse) is 15 feet.
- The distance from the base of the ladder to the tree (one leg of the triangle) is 8 feet.
We need to find the angle θ between the ladder and the tree, which can be represented as follows using the sine function:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this scenario, the "opposite" side is the vertical height of the ladder against the tree, which we can find using the Pythagorean theorem.
First, we will find the height (the opposite side):
\[ \text{height} = \sqrt{(\text{hypotenuse})^2 - (\text{base})^2} \] \[ \text{height} = \sqrt{15^2 - 8^2} = \sqrt{225 - 64} = \sqrt{161} \approx 12.6886 \]
Now, we can calculate the sine of angle θ:
\[ \sin(\theta) = \frac{\text{height}}{\text{hypotenuse}} = \frac{\sqrt{161}}{15} \]
Now we can compute the angle θ using the inverse sine function:
\[ \theta = \sin^{-1}\left(\frac{\sqrt{161}}{15}\right) \]
Calculating this gives:
\[ \theta \approx \sin^{-1}(0.845) \approx 58° \]
Therefore, the approximate angle between the ladder and the tree is 58°.
So the correct response is:
58°
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