Step by step
The question
Question: find the opposite side x of a right triangle with angle 40° and hypotenuse 15.
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
For one angle, every right triangle has the same shape.
Only the size changes.
So each pair of sides always gives the same number.
The calculator knows these numbers.
They are called sin, cos and tan.
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 = 15 and O = x.
So sin 40° is x/15.
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 × sin 40°.
The calculator gives sin 40°: about 0.6428.
Set it to degrees first.
So in every right triangle with a 40° angle, O over H is about 0.6428.
Put the number in.
15 times 0.6428 is about 9.642, with the calculator.
To the nearest tenth, x ≈ 9.6.
5 Check
Let's check: put x back in O over H.
9.6/15 is 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.