Step by step
The question
Question: find the height x of the tree.
The angle of elevation is 45°.
We stand 18 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 45°.
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 adjacent and we want the opposite: A and O.
TOA has them.
So we use tan.
Careful!
The 18 m is along the ground: the adjacent side, not the hypotenuse.
So it is tan, not sin.
3 Write the equation
Now put the numbers in: A = 18 m and O = x.
4 Find x
x is on top, over 18.
To remove the 18 at the bottom, we multiply by 18.
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.
18/18 is 1.
So just x is left.
So x = 18 × tan 45°.
tan 45° is exactly 1.
That one is good to know.
Put the number in.
That is 18.
So x = 18 m.
5 Check
Let's check: put x back in O over A.
We get 1.
That is tan 45° exactly.
So x is right!
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: find x.
Last: check.