Step by step
The question
Question: find the opposite side x of a right triangle with angle 28° and adjacent side 15.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Name the sides
We look from the angle 28°.
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 adjacent and we want the opposite: A and O.
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: A = 15 and O = x.
So tan 28° 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 × tan 28°.
The calculator gives tan 28°: about 0.5317.
Set it to degrees first.
So in every right triangle with a 28° angle, O over A is about 0.5317.
Put the number in.
15 times 0.5317 is about 7.975, with the calculator.
To the nearest tenth, x ≈ 8.0.
5 Check
Let's check: put x back in O over A.
8.0/15 is 0.5333.
tan 28° is about 0.5317.
Very close: the small gap is only from rounding.
So x is right!
6 Pick the choice
Our answer is x ≈ 8.0.
That is choice B.
7 Why not the others
A is 7.0.
Wrong: 15 × sin 28°: sin uses the hypotenuse, not the adjacent side.
C is 13.2.
Wrong: 15 × cos 28°: cos does not give the opposite side.
D is 28.2.
Wrong: 15 ÷ tan 28°: divided instead of multiplying.
Let's look back.
First: name the sides.
Then: choose the ratio.
Then: write the equation.
Then: find x.
Then: check.
Then: pick the choice.
Last: why not the others.