Step by step
The question
Question: find angle B of the triangle with A = 30°, a = 10 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 = 10.
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 10.
The calculator gives sin 30°: exactly 0.5.
8 times 0.5/10 is 0.4.
5 Find B
The sin⁻¹ button turns the ratio back into the angle.
The calculator gives about 23.578.
To the nearest tenth, B ≈ 23.6°.
6 Look for a second triangle
The calculator gives only one angle.
But 180° − 23.6° has the same sine: 156.4°.
It could be B too.
Does it fit?
With angle A, 30°, it makes 186.4°.
That is 180° or more: no room for angle C.
Careful!
Every time we find an angle from its sine, we test 180° minus it too.
Here it does not fit, so there is one triangle.
So there is only one triangle: B ≈ 23.6°.
7 Check
Let's check: put B back.
Both pairs must give the same number.
sin A over a is about 0.0500.
sin B over b: about 0.0500.
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.