Question

you are to construct a decorative wood shelf that has two metal angle supports that fasten to the bottom of the shelf and resemble a right triangle. the shelves are to stick out from the wall 1 foot, and the diagonal distance of the metal piece is 1.25 feet. using the inverse of sine, what is the approximate angle formed where the metal piece rests on the wall? (1 point) 61° 37° 53° 90°

Explanation:

Step1: Identify the right triangle components

In the right triangle, the opposite side to the angle we want to find (let's call it \(\theta\)) is the length the shelf sticks out from the wall, which is \(1\) foot. The hypotenuse (the diagonal metal piece) is \(1.25\) feet. We use the sine function, which is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\).
So, \(\sin(\theta)=\frac{1}{1.25}\)

Step2: Calculate the sine value

\(\frac{1}{1.25} = 0.8\)

Step3: Find the angle using inverse sine

We need to find \(\theta=\sin^{-1}(0.8)\). Using a calculator, \(\sin^{-1}(0.8)\approx53^\circ\) (since \(\sin(53^\circ)\approx0.8\) as a common approximation in right triangles, like the 3-4-5 triangle scaled, where \(\sin(53^\circ)\approx\frac{4}{5} = 0.8\))

Answer:

\(53^\circ\) (corresponding to the option with \(53^\circ\))