Step by step
The question
Question: find side b of the triangle with A = 65°, B = 45° and c = 11.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side c is across from angle C.
Side b, the one we want, is across from angle B.
2 Find the third angle
We need angle C.
The angles of a triangle add to 180°.
So C is 180° take away the other two.
So C = 70°.
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 = c over sin C.
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 = 45°, c = 11 and C = 70°.
5 Find b
b is on top, over sin 45°.
To remove the sin 45° at the bottom, we multiply by sin 45°.
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 45° over sin 45° is 1.
So just b is left.
So b = 11 × sin 45° over sin 70°.
The calculator gives sin 45°: about 0.7071.
And sin 70°: about 0.9397.
Put the numbers in.
That is about 8.277.
To the nearest tenth, b ≈ 8.3.
6 Check
Let's check: both pairs must give the same number.
b over sin 45° is about 11.74.
c over sin 70° is about 11.71.
Almost the same: the gap is only from rounding.
And the bigger angle has the bigger side: 45° is less than 70°, and 8.3 is less than 11.
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.