Step by step
The question
Question: find angle B of the triangle with A = 45°, a = 10 and b = 12.
1 Match sides and angles
Side a is across from angle A: we know both.
They are partners.
Side b is across from angle B, the angle we want.
2 Write the law of sines
A side over the sine of its partner angle is the same for every pair.
To find an angle, we flip both sides: the sines on top.
sin B over b = sin A over a.
3 Put the numbers in
Now put the numbers in: b = 12, A = 45° and a = 10.
4 Find sin B
To remove the 12 at the bottom, we multiply by 12.
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.
12/12 is 1.
So just sin B is left.
So sin B = 12 × sin 45° over 10.
The calculator gives sin 45°: about 0.7071.
12 times 0.7071/10 is about 0.8485.
5 Find B
The sin⁻¹ button turns the ratio back into the angle.
The calculator gives about 58.052.
To the nearest tenth, B ≈ 58.1°.
6 Look for a second triangle
The calculator gives only one angle.
But 180° − 58.1° has the same sine: 121.9°.
It could be B too.
Does it fit?
With angle A, 45°, it makes 166.9°.
That is less than 180°: there is room for angle C.
Careful!
The calculator shows one angle, but here two triangles fit.
Side a can swing in or out and still reach the bottom line.
Here it is: side a swung in.
A second triangle, with B = 121.9°.
So there are two triangles: B ≈ 58.1° or B ≈ 121.9°.
7 Check
Let's check: put B back.
Both pairs must give the same number.
sin A over a is about 0.0707.
sin B over b: about 0.0707 for both angles.
The same, or very close from rounding.
So it is right!
Let's look back.
First: match sides and angles.
Then: write the law of sines.
Then: put the numbers in.
Then: find sin B.
Then: find B.
Then: look for a second triangle.
Last: check.