Step by step
The question
Question: find the height x of the tree.
The angle of elevation is 32°.
We stand 25 m from the tree.
The line of sight goes from the eye to the top of 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 32°.
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
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 adjacent and we want the opposite: A and O.
TOA has them.
So we use tan.
Careful!
The 25 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 = 25 m and O = x.
So tan 32° is x/25.
4 Find x
x is on top, over 25.
To remove the 25 at the bottom, we multiply by 25.
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.
25/25 is 1.
So just x is left.
So x = 25 × tan 32°.
The calculator gives tan 32°: about 0.6249.
Set it to degrees first.
So in every right triangle with a 32° angle, O over A is about 0.6249.
Put the number in.
25 times 0.6249 is about 15.623, with the calculator.
To the nearest tenth, x ≈ 15.6 m.
5 Check
Let's check: put x back in O over A.
15.6/25 is 0.624.
tan 32° is about 0.6249.
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.