Step by step
The question
Question: find the angle of elevation θ.
The tree is 12 m tall.
We stand 20 m from the tree.
The angle of elevation is at the eye.
It goes up from the flat ground to the line of sight.
1 Name the sides
We look from the angle θ.
Every side gets a name from there.
The hypotenuse H is the longest side.
It is across from the right angle.
The opposite side O is across from our angle.
The adjacent side A is next to our angle.
It is not the hypotenuse.
2 Choose the ratio
SOH CAH TOA helps us remember.
Sin is O over H.
Cos is A over H.
Tan is O over A.
We have the opposite and the adjacent: O and A.
TOA has them.
So we use tan.
Careful!
The opposite side is across from the eye.
The adjacent side is along the ground.
3 Write the equation
Now put the numbers in: O = 12 and A = 20.
12 divided by 20 is 0.6.
4 Turn it into the angle
The tan⁻¹ button on the calculator turns a ratio back into the angle.
So θ = tan⁻¹(0.6).
The calculator gives about 30.964.
To the nearest tenth of a degree, θ ≈ 31.0°.
5 Check
Let's check: tan of 31.0° on the calculator.
0.6 again, nearly: the small gap is from rounding the angle.
So θ is right!
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: turn it into the angle.
Last: check.