Step by step
The question
Question: find angle B of the triangle with A = 30°, a = 5 and b = 8.
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 = 8, A = 30° and a = 5.
4 Find sin B
To remove the 8 at the bottom, we multiply by 8.
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.
8/8 is 1.
So just sin B is left.
So sin B = 8 × sin 30° over 5.
The calculator gives sin 30°: exactly 0.5.
8 times 0.5/5 is 0.8.
5 Find B
The sin⁻¹ button turns the ratio back into the angle.
The calculator gives about 53.13.
To the nearest tenth, B ≈ 53.1°.
6 Look for a second triangle
The calculator gives only one angle.
But 180° − 53.1° has the same sine: 126.9°.
It could be B too.
Does it fit?
With angle A, 30°, it makes 156.9°.
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 = 126.9°.
So there are two triangles: B ≈ 53.1° or B ≈ 126.9°.
7 Check
Let's check: put B back.
Both pairs must give the same number.
sin A over a is about 0.1000.
sin B over b: about 0.1000 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.