To find the angle between the height of the tree and the owl's line of sight using the inverse sine function, we can use the following steps:
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Identify the components of the triangle:
- The horizontal distance from you to the base of the tree is 5 feet.
- The diagonal distance to the owl (the hypotenuse) is 14 feet.
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Find the height of the owl in the tree: To find the height (opposite side) of the angle we are trying to find, we can use the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] where:
- \(c = 14\) feet (hypotenuse),
- \(a = 5\) feet (adjacent side),
- \(b\) is the height of the owl (opposite side).
Rearranging gives: \[ b^2 = c^2 - a^2 \] \[ b^2 = 14^2 - 5^2 \] \[ b^2 = 196 - 25 = 171 \] \[ b = \sqrt{171} \approx 13.08 \text{ feet} \]
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Find the angle using the sine function: We use the sine function to find the angle \( \theta \): \[ \sin(\theta) = \frac{ \text{opposite side}}{\text{hypotenuse}} = \frac{b}{c} = \frac{13.08}{14} \] \[ \sin(\theta) \approx 0.9357 \]
Now we calculate the angle: \[ \theta \approx \sin^{-1}(0.9357) \approx 68.9 \text{ degrees} \]
From the options given:
- \(29.9\) Degrees
- \(15.1\) Degrees
- \(69.1\) Degrees
- \(19.7\) Degrees
The closest option to our calculated angle of \(68.9\) degrees is \(69.1\) degrees. Thus, the angle between the height of the tree and the owl's line of sight looking down at you is 69.1 Degrees.