We are told that a roof has a slope of 2 3 and an angle of elevation θ with the horizontal. Let's imagine this triangular figure with its height.
We want to find the connection between the slope and the angle of elevation θ. Since it is given a positive slope, we can consider only the right-hand side of the triangular roof.
Now, we will recall that slope is a ratio of rise over run. With this in mind, we will write the opposite side of θ as rise and the adjacent side of θ as run.
Slope: rise/run ⇒ 2/3
Let's show this slope in the figure! But first, we need to expand our ratio with a factor of k because we do not know the actual side lengths.
Notice that we can relate θ to the values of 2k and 3k by considering some trigonometric functions. Since we have expressions for the opposite side and adjacent side, we can simply choose the tangent function.
tan( θ)=Opposite Side/Adjacent Side ⇒ 2k/3k
We can conclude that the tangent value of the angle of elevation θ is equal to the slope of 2 3.
tan( θ)= 2k/3k=2/3
We will now use the inverse tangent function to find the value of θ.
tan( θ)=2/3 ⇔ tan^(-1)(2/3)= θ
To do so we will use a calculator. We need to first push MODE in our calculator and change Radian to Degree.
Then, we can push 2ND followed by TAN, introduce the value 2 3, and push ENTER.
The inverse tangent of 2 3 is about 33.7^(∘), which corresponds to the value of θ.
θ ≈ 33.7 ^(∘)
