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..

Answer 1

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 inverse sine function. We have:

- The distance from you to the house is \( a = 12 \) feet (the horizontal distance).
- The diagonal distance between you and the cat is \( c = 18 \) feet (the hypotenuse of the right triangle formed).

To find the height of the house where the cat is sitting, we first need to calculate the height \( b \) using the Pythagorean theorem:

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

Substituting the known values:

\[
18^2 = 12^2 + b^2
\]
\[
324 = 144 + b^2
\]
\[
b^2 = 324 - 144
\]
\[
b^2 = 180
\]
\[
b = \sqrt{180} \approx 13.42 \text{ feet}
\]

Next, we will use the heights and the horizontal distance to find the angle \( \theta \). The sine of the angle \( \theta \) is given by the ratio of the opposite side (height of the house) to the hypotenuse (diagonal distance):

\[
\sin(\theta) = \frac{b}{c} = \frac{b}{18}
\]
Substituting the height \( b \):

\[
\sin(\theta) = \frac{\sqrt{180}}{18}
\]

Now we calculate:

\[
\sin(\theta) = \frac{13.42}{18} \approx 0.746
\]

Now we use the inverse sine (arcsin) function to find \( \theta \):

\[
\theta = \arcsin(0.746)
\]

Using a calculator:

\[
\theta \approx 48.5 \text{ degrees}
\]

Rounding to the nearest whole degree:

\[
\theta \approx 49 \text{ degrees}
\]

So the angle between the side of the house and the cat's line of sight looking down at you is approximately **49 degrees**.