Question

A cat climbs on top of a house and looks down at you. You are standing 12 feet away from the house and there is an 18-foot diagonal distance between you and the cat. Using the inverse of sine function, find the angle between the side of the house and the cat's line of sight looking down at you. Round your answer to the nearest whole degree.(1 point)
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Answer 1

To solve for the angle between the side of the house and the cat's line of sight, we can use the inverse sine function. Let's denote the following:

- \( a \): the height of the house (the vertical leg of the triangle)
- \( b \): the distance from the house to you (the horizontal leg of the triangle), which is 12 feet
- \( c \): the diagonal distance from the cat to you, which is 18 feet

We need to find the angle \( \theta \) at the base of the triangle formed by the cat, the top of the house, and you.

Using the sine function, we have:

\[
\sin(\theta) = \frac{a}{c}
\]

However, we don't have the height \( a \) of the house directly. We can find it by using the Pythagorean theorem:

\[
c^2 = a^2 + b^2
\]

Substituting the values we have:

\[
18^2 = a^2 + 12^2
\]

This simplifies to:

\[
324 = a^2 + 144
\]

Subtracting 144 from both sides:

\[
a^2 = 324 - 144 = 180
\]

Taking the square root gives:

\[
a = \sqrt{180} \approx 13.42 \text{ feet}
\]

Now we can find \( \theta \) using:

\[
\sin(\theta) = \frac{a}{c} = \frac{13.42}{18}
\]

Calculating this gives:

\[
\sin(\theta) \approx 0.7467
\]

Now, to find \( \theta \):

\[
\theta = \sin^{-1}(0.7467) \approx 48.8^\circ
\]

Rounding to the nearest whole degree, we find:

\[
\theta \approx 49^\circ
\]

Thus, the angle between the side of the house and the cat's line of sight looking down at you is \( \approx 49^\circ \).