Question
An owl is in a tree looking down at you. You are standing 5 feet away from the tree and there is a 14 foot diagonal distance between you and the owl.
Using the inverse of sine function, find the angle between the height of the tree and the owl's line of sight looking down at you.
Using the inverse of sine function, find the angle between the height of the tree and the owl's line of sight looking down at you.
Answers
GPT-4o mini
To find the angle between the height of the tree and the owl's line of sight, we can use the inverse sine function in conjunction with the right triangle formed by the distance from the base of the tree to your position, the height of the owl, and the diagonal distance from you to the owl.
Let:
- \( d \) = 5 feet (the horizontal distance from you to the tree)
- \( h \) = the height of the owl in feet (we'll derive this from the triangle)
- \( L \) = 14 feet (the diagonal distance from you to the owl)
In this triangle, we can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]
where:
- \( \theta \) is the angle we want to find (the angle between the height of the tree and the owl's line of sight).
- \( h \) is the height of the owl above the ground.
To find \( h \), we can use the Pythagorean theorem since we have a right triangle:
\[
L^2 = h^2 + d^2
\]
Substituting the values for \( L \) and \( d \):
\[
14^2 = h^2 + 5^2
\]
\[
196 = h^2 + 25
\]
\[
h^2 = 196 - 25
\]
\[
h^2 = 171
\]
\[
h = \sqrt{171}
\]
Now, we can calculate \( \sin(\theta) \):
\[
\sin(\theta) = \frac{h}{L} = \frac{\sqrt{171}}{14}
\]
Next, we can find \( \theta \) using the inverse sine function:
\[
\theta = \arcsin\left(\frac{\sqrt{171}}{14}\right)
\]
Calculating it:
1. First, estimate \( \sqrt{171} \approx 13.0384 \).
\[
\sin(\theta) \approx \frac{13.0384}{14} \approx 0.9313
\]
2. Now find \( \theta \):
\[
\theta \approx \arcsin(0.9313) \approx 67.38^{\circ}
\]
Thus, the angle between the height of the tree and the owl's line of sight looking down at you is approximately \( 67.38^{\circ} \).
Let:
- \( d \) = 5 feet (the horizontal distance from you to the tree)
- \( h \) = the height of the owl in feet (we'll derive this from the triangle)
- \( L \) = 14 feet (the diagonal distance from you to the owl)
In this triangle, we can use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]
where:
- \( \theta \) is the angle we want to find (the angle between the height of the tree and the owl's line of sight).
- \( h \) is the height of the owl above the ground.
To find \( h \), we can use the Pythagorean theorem since we have a right triangle:
\[
L^2 = h^2 + d^2
\]
Substituting the values for \( L \) and \( d \):
\[
14^2 = h^2 + 5^2
\]
\[
196 = h^2 + 25
\]
\[
h^2 = 196 - 25
\]
\[
h^2 = 171
\]
\[
h = \sqrt{171}
\]
Now, we can calculate \( \sin(\theta) \):
\[
\sin(\theta) = \frac{h}{L} = \frac{\sqrt{171}}{14}
\]
Next, we can find \( \theta \) using the inverse sine function:
\[
\theta = \arcsin\left(\frac{\sqrt{171}}{14}\right)
\]
Calculating it:
1. First, estimate \( \sqrt{171} \approx 13.0384 \).
\[
\sin(\theta) \approx \frac{13.0384}{14} \approx 0.9313
\]
2. Now find \( \theta \):
\[
\theta \approx \arcsin(0.9313) \approx 67.38^{\circ}
\]
Thus, the angle between the height of the tree and the owl's line of sight looking down at you is approximately \( 67.38^{\circ} \).