Step by step
The question
Question: find side b of the triangle with B = 40°, C = 30° and a = 15 m.
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 Find the third angle
We need angle A.
The angles of a triangle add to 180°.
So A is 180° take away the other two.
So A = 110°.
3 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.
Careful!
Each side goes with the angle across from it, never with an angle next to it.
4 Put the numbers in
Now put the numbers in: B = 40°, a = 15 and A = 110°.
5 Find b
b is on top, over sin 40°.
To remove the sin 40° at the bottom, we multiply by sin 40°.
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 40° over sin 40° is 1.
So just b is left.
So b = 15 × sin 40° over sin 110°.
The calculator gives sin 40°: about 0.6428.
And sin 110°: about 0.9397.
Put the numbers in.
That is about 10.261.
To the nearest tenth, b ≈ 10.3 m.
6 Check
Let's check: both pairs must give the same number.
b over sin 40° is about 16.02.
a over sin 110° is about 15.96.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 40° is less than 110°, and 10.3 is less than 15.
So it is right!
Let's look back.
First: match sides and angles.
Then: find the third angle.
Then: write the law of sines.
Then: put the numbers in.
Then: find b.
Last: check.