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°.