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 solve the problem, we can use the information given to determine the angle between the height of the tree and the owl's line of sight using the inverse sine function.
Let's denote:
- \( h \) = height of the owl in the tree (which we need to find).
- \( d \) = horizontal distance from you to the tree = 5 feet.
- \( L \) = diagonal distance from you to the owl (the line of sight) = 14 feet.
From the right triangle formed by the tree, you, and the owl, we can use the Pythagorean theorem to find the height of the owl:
\[
L^2 = h^2 + d^2
\]
Substituting the known values:
\[
14^2 = h^2 + 5^2
\]
\[
196 = h^2 + 25
\]
Subtracting 25 from both sides gives:
\[
h^2 = 196 - 25
\]
\[
h^2 = 171
\]
Now taking the square root of both sides gives:
\[
h = \sqrt{171} \approx 13.08 \text{ feet}
\]
Now, to find the angle \( \theta \) between the height of the tree (which is the vertical side) and the owl's line of sight (the hypotenuse), we use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]
Substituting \( h \) and \( L \):
\[
\sin(\theta) = \frac{\sqrt{171}}{14}
\]
Next, we will calculate the angle \( \theta \) using the inverse sine function:
\[
\theta = \sin^{-1}\left(\frac{\sqrt{171}}{14}\right)
\]
Calculating \( \frac{\sqrt{171}}{14} \):
1. Calculate \( \sqrt{171} \approx 13.08\)
2. Now find the ratio:
\[
\frac{13.08}{14} \approx 0.9343
\]
Next, use the inverse sine function:
\[
\theta \approx \sin^{-1}(0.9343)
\]
Calculating this gives approximately:
\[
\theta \approx 68.9^\circ
\]
Therefore, the angle between the height of the tree and the owl's line of sight is approximately \( 68.9^\circ \).
Let's denote:
- \( h \) = height of the owl in the tree (which we need to find).
- \( d \) = horizontal distance from you to the tree = 5 feet.
- \( L \) = diagonal distance from you to the owl (the line of sight) = 14 feet.
From the right triangle formed by the tree, you, and the owl, we can use the Pythagorean theorem to find the height of the owl:
\[
L^2 = h^2 + d^2
\]
Substituting the known values:
\[
14^2 = h^2 + 5^2
\]
\[
196 = h^2 + 25
\]
Subtracting 25 from both sides gives:
\[
h^2 = 196 - 25
\]
\[
h^2 = 171
\]
Now taking the square root of both sides gives:
\[
h = \sqrt{171} \approx 13.08 \text{ feet}
\]
Now, to find the angle \( \theta \) between the height of the tree (which is the vertical side) and the owl's line of sight (the hypotenuse), we use the sine function:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{L}
\]
Substituting \( h \) and \( L \):
\[
\sin(\theta) = \frac{\sqrt{171}}{14}
\]
Next, we will calculate the angle \( \theta \) using the inverse sine function:
\[
\theta = \sin^{-1}\left(\frac{\sqrt{171}}{14}\right)
\]
Calculating \( \frac{\sqrt{171}}{14} \):
1. Calculate \( \sqrt{171} \approx 13.08\)
2. Now find the ratio:
\[
\frac{13.08}{14} \approx 0.9343
\]
Next, use the inverse sine function:
\[
\theta \approx \sin^{-1}(0.9343)
\]
Calculating this gives approximately:
\[
\theta \approx 68.9^\circ
\]
Therefore, the angle between the height of the tree and the owl's line of sight is approximately \( 68.9^\circ \).