Question

Two birds sit at the top of two different trees. The distance between the first bird and a birdwatcher on the ground is 29.7 feet. The distance between the birdwatcher and the second bird is 42.1 feet.

What is the angle measure, or angle of depression, between this bird and the birdwatcher?

Two trees that are 29 point 7 feet tall with a bird on top of each. Distance between birdwatcher and first bird is 42 point 1 feet. Angle of depression between birdwatcher and first bird is unknown.
© 2011 Jupiter Images Corporation

45.1°
44.9°
54.8°
35.2°

Answer 1

To find the angle of depression from the birdwatcher to the first bird, we can use trigonometry. The situation describes a right triangle where:

- The height of the tree (which is also the height of the bird) is 29.7 feet (opposite side).
- The distance from the birdwatcher to the base of the tree (the same as the horizontal distance from the birdwatcher to the foot of the tree) can be calculated using the Pythagorean theorem.

We have two sides of the right triangle:
1. The height of the tree (opposite side, \( h = 29.7 \) feet).
2. The hypotenuse distance to the birdwatcher from the first bird (which is \( d = 42.1 \) feet).

We want to find the angle of depression \( \theta \), which can be calculated using the sine function:

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

Substituting the values:
\[
\sin(\theta) = \frac{29.7}{42.1}
\]

Calculating:
\[
\sin(\theta) = 0.7052
\]

To find the angle \( \theta \), take the inverse sine (\(\sin^{-1}\)):
\[
\theta = \sin^{-1}(0.7052)
\]

Calculating this gives:
\[
\theta \approx 44.9°
\]

Therefore, the angle measure, or angle of depression, between the bird and the birdwatcher is approximately **44.9°**.