Step by step
The question
Question: find the opposite side x of a right triangle with angle 40° and hypotenuse 10.
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 hypotenuse and we want the opposite: H and O.
SOH has them.
So we use sin.
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 = 10 and O = x.
4 Find x
x is on top, over 10.
To remove the 10 at the bottom, we multiply by 10.
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.
10/10 is 1.
So just x is left.
So x = 10 × sin 40°.
The calculator gives sin 40°: about 0.6428.
Put the number in.
10 times 0.6428 is about 6.428.
To the nearest tenth, x ≈ 6.4.
5 Check
Let's check: put x back in O over H.
We get 0.64.
sin 40° is about 0.6428.
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.