Step by step
The question
Question: find the adjacent side x of a right triangle with angle 40° and opposite side 5.
1 Name the sides
We look from the angle 40°.
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.
Careful!
From the other angle, the opposite and adjacent sides would swap.
Always name them from the angle we use.
3 Write the equation
Now put the numbers in: O = 5 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 40° = 5.
To remove times tan 40°, 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 40°: about 0.8391.
Put the number in.
5 divided by 0.8391 is about 5.959, with the calculator.
To the nearest tenth, x ≈ 6.0.
5 Check
Let's check: put x back in O over A.
We get 0.8333.
tan 40° is about 0.8391.
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.