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)

Question

Responses

58°
58°

25°
25°

90°
90°

32°

Answer 1

To find the angle between the ladder and the tree using the inverse sine (arcsin), we can follow these steps:

1. Identify the lengths involved in the triangle:

   • The ladder (hypotenuse) = 15 feet
   • The distance from the base of the ladder to the tree (adjacent side) = 8 feet

2. We need to find the opposite side (the height at which the ladder touches the tree) using the Pythagorean theorem. However, since you asked for the angle, we can also use the sine function directly.

3. The sine of the angle θ between the ladder and the ground can be found using the formula:

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

4. We need to determine the height (opposite side) using the Pythagorean theorem:

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

5. Now we can use the ratios to find the angle. From the correct triangle:

   \[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{8}{15} \]

6. To find the angle θ:

   \[ \theta = \cos^{-1}\left(\frac{8}{15}\right) \]

7. Therefore, to find the original angle between the ladder and the vertical (tree), we can use:

   \[ \theta = 90° - \cos^{-1}\left(\frac{8}{15}\right) \]

Alternatively, you can also calculate:

\[ \sin(\theta) = \frac{12.688}{15} \]

\[ \theta = \sin^{-1}\left(\frac{12.6886}{15}\right) \approx 58° \]

Thus, the angle between the ladder and the tree is approximately 58°.