To find the angle between the side of the house and the cat's line of sight looking down at you, we can use the definition of the sine function in the context of a right triangle.
The situation can be described with the following information:
- The horizontal distance from you to the base of the house (adjacent side) = 12 feet.
- The diagonal distance (hypotenuse) from the cat to you = 18 feet.
We can use the sine function to find the angle \( \theta \) that we are looking for:
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this case, the "opposite" side is the height of the house (which we do not know), and the "hypotenuse" is 18 feet. However, we can find the height using the Pythagorean theorem, where you treat the adjacent, opposite, and hypotenuse sides of the triangle formed by the vertical distance (height of the cat) to the ground.
First, we need to find the height (opposite side) using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{adjacent}^2 + \text{opposite}^2 \]
Substituting in the known values:
\[ 18^2 = 12^2 + h^2 \] \[ 324 = 144 + h^2 \] \[ h^2 = 324 - 144 \] \[ h^2 = 180 \] \[ h = \sqrt{180} \approx 13.42 \text{ feet} \]
Now that we have the height, we can use it to find the angle \( \theta \):
Using the sine function, we can express it as:
\[ \sin(\theta) = \frac{h}{\text{hypotenuse}} = \frac{h}{18} \]
Substituting in the value of \( h \):
\[ \sin(\theta) = \frac{13.42}{18} \] \[ \sin(\theta) \approx 0.746 \]
Now, we can find the angle \( \theta \) using the inverse sine function:
\[ \theta = \sin^{-1}(0.746) \] \[ \theta \approx 48.5^\circ \]
Rounding to the nearest whole degree:
\[ \theta \approx 49^\circ \]
Thus, the angle between the side of the house and the cat's line of sight looking down at you is approximately 49 degrees.
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