Step by step
The question
Question: find the opposite side x of a right triangle with angle 30° and hypotenuse 12.
1 Name the sides
We look from the angle 30°.
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 = 12 and O = x.
4 Find x
x is on top, over 12.
To remove the 12 at the bottom, we multiply by 12.
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.
12/12 is 1.
So just x is left.
So x = 12 × sin 30°.
sin 30° is exactly 0.5.
That one is good to know.
Put the number in.
That is 6.
So x = 6.
5 Check
Let's check: put x back in O over H.
We get 0.5.
That is sin 30° exactly.
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.