Question

A cat climbs onto a roof 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 cats line of sigh 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, we can use the sine function.

We have:
- The opposite side (distance from the base of the house to the cat) = height of the cat from the ground (let's denote it as \( h \)).
- The hypotenuse (the diagonal distance from you to the cat) = 18 feet.
- The adjacent side (distance from you to the house) = 12 feet.

Using the sine function:

\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{18}
\]

To find \( h \), we can use the Pythagorean theorem:

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

Calculating \( 12^2 \) and \( 18^2 \):

\[
h^2 + 144 = 324
\]

Subtracting 144 from both sides:

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

Taking the square root of both sides:

\[
h = \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} \approx 13.42 \text{ feet}
\]

Now, we can use this height to find the angle \( \theta \):

\[
\sin(\theta) = \frac{h}{18} = \frac{6\sqrt{5}}{18} = \frac{\sqrt{5}}{3}
\]

Now, we calculate \( \theta \):

\[
\theta = \arcsin\left(\frac{\sqrt{5}}{3}\right)
\]

Using a calculator to find the arcsine value, we calculate \( \frac{\sqrt{5}}{3} \approx 0.745 \).

Now finding \( \theta \):

\[
\theta \approx \arcsin(0.745) \approx 48.61^\circ
\]

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 \( \boxed{49} \) degrees.