Step by step
The question
Question: find the distance x to the boat.
The angle of depression is 25°.
The lighthouse is 45 m tall.
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 25° too.
Alternate angles are equal.
2 Name the sides
We look from the angle 25°.
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 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.
Inside the triangle at the top, the angle is 65°.
4 Write the equation
Now put the numbers in: O = 45 m and A = x.
5 Find x
x is at the bottom.
First 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 25° = 45.
To remove times tan 25°, 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 25°: about 0.4663.
Put the number in.
45 divided by 0.4663 is about 96.504, with the calculator.
To the nearest tenth, x ≈ 96.5 m.
6 Check
Let's check: put x back in O over A.
We get 0.4663.
That is tan 25° exactly.
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.