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. Use 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.
Answers
GPT-4o mini
To solve this problem, we need to find the angle between the side of the house (which we can think of as being vertical, or at a 90-degree angle) and the line of sight from the cat to you.
We have the following information:
- The distance from the base of the house to where you are standing (adjacent side) = 12 feet.
- The diagonal distance (hypotenuse) from the cat to you = 18 feet.
We can use the inverse sine function (arcsin) to find the angle. First, we want to find the length of the opposite side, which is the height of the house (the height at which the cat is located). We can use the Pythagorean theorem to find this height.
Let \( h \) be the height of the house (the opposite side). According to the Pythagorean theorem:
\[
h^2 + (12)^2 = (18)^2
\]
Calculating \( (12)^2 \) and \( (18)^2 \):
\[
h^2 + 144 = 324
\]
Now, isolating \( h^2 \):
\[
h^2 = 324 - 144 = 180
\]
Taking the square root to find \( h \):
\[
h = \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} \approx 13.42 \text{ feet}
\]
Now, we can find the angle \( \theta \) between the side of the house and the cat's line of sight using the sine function, which is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{18}
\]
Substituting the values we have:
\[
\sin(\theta) = \frac{6\sqrt{5}}{18} = \frac{\sqrt{5}}{3}
\]
To find the angle \( \theta \):
\[
\theta = \arcsin\left(\frac{\sqrt{5}}{3}\right)
\]
Now, we can calculate \( \frac{\sqrt{5}}{3} \):
Calculating \( \sqrt{5} \approx 2.236 \).
Thus,
\[
\frac{\sqrt{5}}{3} \approx \frac{2.236}{3} \approx 0.745
\]
Now using a calculator to find \( \theta \):
\[
\theta \approx \arcsin(0.745) \approx 48.5 \text{ degrees}
\]
Rounding to the nearest whole degree, we find:
\[
\theta \approx 49 \text{ degrees}
\]
Therefore, the angle between the side of the house and the cat's line of sight is approximately **49 degrees**.
We have the following information:
- The distance from the base of the house to where you are standing (adjacent side) = 12 feet.
- The diagonal distance (hypotenuse) from the cat to you = 18 feet.
We can use the inverse sine function (arcsin) to find the angle. First, we want to find the length of the opposite side, which is the height of the house (the height at which the cat is located). We can use the Pythagorean theorem to find this height.
Let \( h \) be the height of the house (the opposite side). According to the Pythagorean theorem:
\[
h^2 + (12)^2 = (18)^2
\]
Calculating \( (12)^2 \) and \( (18)^2 \):
\[
h^2 + 144 = 324
\]
Now, isolating \( h^2 \):
\[
h^2 = 324 - 144 = 180
\]
Taking the square root to find \( h \):
\[
h = \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} \approx 13.42 \text{ feet}
\]
Now, we can find the angle \( \theta \) between the side of the house and the cat's line of sight using the sine function, which is defined as:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{18}
\]
Substituting the values we have:
\[
\sin(\theta) = \frac{6\sqrt{5}}{18} = \frac{\sqrt{5}}{3}
\]
To find the angle \( \theta \):
\[
\theta = \arcsin\left(\frac{\sqrt{5}}{3}\right)
\]
Now, we can calculate \( \frac{\sqrt{5}}{3} \):
Calculating \( \sqrt{5} \approx 2.236 \).
Thus,
\[
\frac{\sqrt{5}}{3} \approx \frac{2.236}{3} \approx 0.745
\]
Now using a calculator to find \( \theta \):
\[
\theta \approx \arcsin(0.745) \approx 48.5 \text{ degrees}
\]
Rounding to the nearest whole degree, we find:
\[
\theta \approx 49 \text{ degrees}
\]
Therefore, the angle between the side of the house and the cat's line of sight is approximately **49 degrees**.