Step by step
The question
Question: find the adjacent side x of a right triangle with angle 70° and hypotenuse 15 cm.
1 Name the sides
We look from the angle 70°.
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 adjacent: H and A.
CAH has them.
So we use cos.
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: H = 15 cm and A = x.
4 Find x
x is on top, over 15.
To remove the 15 at the bottom, we multiply by 15.
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.
15/15 is 1.
So just x is left.
So x = 15 × cos 70°.
The calculator gives cos 70°: about 0.342.
Put the number in.
15 times 0.342 is about 5.13.
To the nearest tenth, x ≈ 5.1 cm.
5 Check
Let's check: put x back in A over H.
We get 0.34.
cos 70° is about 0.342.
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.