Step by step
The question
Question: find the distance x to the boat.
The angle of depression is 28°.
The cliff is 65 m tall.
1 Move the angle down
The line of sight goes from the top down to the boat.
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: both are flat, so they never meet.
The line of sight crosses both.
The angle at the top and the angle at the boat sit at the two ends of a Z.
Angles at the ends of a Z between parallel lines are always equal: alternate angles.
So the angle at the boat is 28° too.
Alternate angles are equal.
2 Name the sides
We look from the angle 28°.
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
For one angle, every right triangle has the same shape.
Only the size changes.
So each pair of sides always gives the same number.
The calculator knows these numbers.
They are called sin, cos and tan.
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 we want 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.
The angle inside the triangle at the top is a different angle.
4 Write the equation
Now put the numbers in: O = 65 m and A = x.
So tan 28° is 65 over x.
5 Find x
x is at the bottom.
To get it off the bottom, we multiply by x.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
x/x is 1.
Only the top is left.
Now we have x × tan 28° = 65.
To remove times tan 28°, we divide by it.
We do it on both sides, so they stay equal.
It divides by itself: 1.
So just x is left.
The calculator gives tan 28°: about 0.5317.
Set it to degrees first.
So in every right triangle with a 28° angle, O over A is about 0.5317.
Put the number in.
65 divided by 0.5317 is about 122.249, with the calculator.
To the nearest tenth, x ≈ 122.2 m.
6 Check
Let's check: put x back in O over A.
65/122.2 is 0.5319.
tan 28° is about 0.5317.
Very close: the small gap is only from rounding.
So x is right!
Let's look back.
First: move the angle down.
Then: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: find x.
Last: check.