To find the angle between the ladder and the tree using the inverse sine function, we can use the formula:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this scenario, the opposite side is the distance from the base of the ladder to the point where it touches the tree, and the hypotenuse is the length of the ladder.
- First, we need to determine the height at which the ladder touches the tree using the Pythagorean theorem. We know:
- The length of the ladder (hypotenuse) = 33 feet
- The distance from the base of the ladder to the tree (adjacent) = 9 feet
Using the Pythagorean theorem: \[ \text{Height}^2 + \text{Base}^2 = \text{Hypotenuse}^2 \] \[ \text{Height}^2 + 9^2 = 33^2 \] \[ \text{Height}^2 + 81 = 1089 \] \[ \text{Height}^2 = 1089 - 81 = 1008 \] \[ \text{Height} = \sqrt{1008} \approx 31.76 \text{ feet} (not actually needed for finding the angle) \]
- Now, we can calculate the angle using the inverse sine: \[ \sin(\theta) = \frac{\text{opposite (height)}}{\text{hypotenuse (ladder length)}} \] So, we can use: \[ \sin(\theta) = \frac{31.76}{33} \]
However, we can also calculate the angle more directly as: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{9}{33} \]
Then, \[ \theta = \cos^{-1}\left(\frac{9}{33}\right) \]
Now calculating: \[ \frac{9}{33} = 0.2727 \] \[ \theta = \cos^{-1}(0.2727) \approx 74.74° \]
- Since we need to find the angle between the tree and the ladder, we need to subtract from 90°: \[ \text{Angle with the tree} = 90° - \theta \] \[ \approx 90° - 74.74° \approx 15.26° \]
Therefore, the closest available option to approximately 15° is 16°.
The approximate answer for the angle between the ladder and the tree is 16°.