Question

inverse sine quick check
romeo owns a business that puts up and takes down holiday lights. he is working on a house and places the base of his 20 - foot ladder at a 4 - foot distance from the house. using the inverse of sine, what is the approximate angle formed where the ladder rests on the house? (1 point)
○ 90°
○ 12°
○ 78°
○ 34°

Explanation:

Step1: Identify the triangle components

We have a right triangle where the ladder is the hypotenuse (\(c = 20\) ft), the distance from the house is the adjacent side (\(a = 4\) ft)? Wait, no, wait. Wait, the height on the house would be the opposite side. Wait, actually, the problem says "places the base of his 20 - foot ladder at a 4 - foot distance from the house". So the ladder is hypotenuse (\(c = 20\)), the distance from the house (base) is \(b = 4\), and the height on the house is \(a\). We need to find the angle between the ladder and the house, which is the angle between the hypotenuse and the opposite side? Wait, no. Wait, the angle formed where the ladder rests on the house: so the ladder is the hypotenuse, the house is the vertical side (opposite side to the angle at the base), and the ground is the horizontal side (adjacent side to the angle at the base). Wait, no, the angle where the ladder rests on the house: so the ladder (hypotenuse) and the house (vertical side) form an angle. Let's denote the angle at the house as \(\theta\). Then, in the right triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), but wait, no. Wait, the adjacent side to \(\theta\) would be the height on the house, and the opposite side would be the distance from the house (4 ft). Wait, no, let's draw it: house is vertical, ground is horizontal, ladder is hypotenuse. The base of the ladder is 4 ft from the house (so horizontal distance \(x = 4\) ft), ladder length \(L = 20\) ft. The angle between the ladder and the house (let's call it \(\theta\)): in this case, the side opposite to \(\theta\) is the horizontal distance (4 ft), and the hypotenuse is 20 ft. So \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{20}=0.2\). Then \(\theta=\sin^{-1}(0.2)\).

Step2: Calculate the inverse sine

We calculate \(\theta=\sin^{-1}(0.2)\). Using a calculator, \(\sin^{-1}(0.2)\approx11.54^{\circ}\), which is approximately \(12^{\circ}\). Wait, but the options are \(90^{\circ}\), \(12^{\circ}\), \(78^{\circ}\), \(34^{\circ}\). Wait, maybe I mixed up the angle. Wait, maybe the angle at the ground? Wait, no, the problem says "the angle formed where the ladder rests on the house". Let's re - examine. If the ladder is 20 ft, base is 4 ft from the house. So the angle between the ladder and the house: let's consider the right triangle with hypotenuse 20, opposite side (to the angle at the house) is 4. So \(\sin\theta=\frac{4}{20}=0.2\), so \(\theta=\sin^{-1}(0.2)\approx12^{\circ}\).

Answer:

\(12^{\circ}\) (the option with \(12^{\circ}\))