Step by step
The question
Question: find the distance x to the tree.
The angle of elevation is 50°.
The tree is 20 m tall.
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 50°.
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 we want the adjacent: O and A.
TOA has them.
So we use tan.
3 Write the equation
Now put the numbers in: O = 20 m and A = x.
4 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 50° = 20.
To remove times tan 50°, 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 50°: about 1.1918.
Put the number in.
20 divided by 1.1918 is about 16.781, with the calculator.
To the nearest tenth, x ≈ 16.8 m.
5 Check
Let's check: put x back in O over A.
We get 1.1905.
tan 50° is about 1.1918.
Very close: the small gap is only from rounding.
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.