Step by step
The question
Question: find side b of the triangle with A = 40°, B = 65° and a = 12.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side a is across from angle A.
Side b, the one we want, is across from angle B.
2 Write the law of sines
In every triangle, a side over the sine of its partner angle gives the same number, for all three pairs.
We use the pair we want and the pair we know: b over sin B = a over sin A.
Why is it true?
Draw the height h from corner C, straight down to the side across from it.
It makes two right triangles.
In one, h is b times sin A.
In the other, h is a times sin B.
Both are h, so they are equal.
Divide both sides by sin B times sin A.
Each side is left over the sine of its partner angle.
Careful!
Each side goes with the angle across from it, never with an angle next to it.
3 Put the numbers in
Now put the numbers in: B = 65°, a = 12 and A = 40°.
4 Find b
b is on top, over sin 65°.
To remove the sin 65° at the bottom, we multiply by sin 65°.
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.
sin 65° over sin 65° is 1.
So just b is left.
So b = 12 × sin 65° over sin 40°.
The calculator gives sin 65°: about 0.9063.
And sin 40°: about 0.6428.
Put the numbers in.
That is about 16.92.
To the nearest tenth, b ≈ 16.9.
5 Check
Let's check: both pairs must give the same number.
b over sin 65° is about 18.65.
a over sin 40° is about 18.67.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: in every triangle the biggest angle faces the longest side.
65° is more than 40°, and 16.9 is more than 12.
So it is right!
6 Pick the choice
Our answer is b ≈ 16.9.
That is choice C.
7 Why not the others
A is 8.5.
Wrong: upside down: b/sin B = a/sin A.
B is 10.9.
Wrong: 12 × sin 65°: forgot to divide by sin 40°.
D is 19.5.
Wrong: used the angles, not their sines.
Let's look back.
First: match sides and angles.
Then: write the law of sines.
Then: put the numbers in.
Then: find b.
Then: check.
Then: pick the choice.
Last: why not the others.