Step by step
The question
Question: find the height x of the tree.
The angle of elevation is 55°.
The wire is 25 m long.
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 55°.
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 hypotenuse and we want the opposite: H and O.
SOH has them.
So we use sin.
Careful!
The 25 m is the slanted side: the hypotenuse, not the ground.
So it is sin, not tan.
3 Write the equation
Now put the numbers in: H = 25 m and O = x.
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 × sin 55°.
The calculator gives sin 55°: about 0.8192.
Put the number in.
25 times 0.8192 is about 20.48.
To the nearest tenth, x ≈ 20.5 m.
5 Check
Let's check: put x back in O over H.
We get 0.82.
sin 55° is about 0.8192.
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.