Step by step
The question
Question: find the angle of depression θ.
The cliff is 40 m tall.
We stand 100 m from the cliff.
1 Move the angle down
The angle of depression is at the top.
It goes down from the flat dashed line to the line of sight.
The flat line and the ground are parallel.
The line of sight crosses both.
So the angle at the boat is the same: alternate angles, a Z shape.
So the angle at the boat is θ too.
Alternate angles are equal.
2 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.
3 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 angle of depression is between the flat line and the line of sight.
4 Write the equation
Now put the numbers in: O = 40 and A = 100.
40 divided by 100 is 0.4.
5 Turn it into the angle
The tan⁻¹ button on the calculator turns a ratio back into the angle.
So θ = tan⁻¹(0.4).
The calculator gives about 21.801.
To the nearest tenth of a degree, θ ≈ 21.8°.
6 Check
Let's check: tan of 21.8° on the calculator.
That is 0.4 again.
So θ is right!
Let's look back.
First: move the angle down.
Then: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: turn it into the angle.
Last: check.