Step by step
The question
Question: find angle B of the triangle with A = 50°, a = 12 and b = 9.
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 = 9, A = 50° and a = 12.
4 Find sin B
To remove the 9 at the bottom, we multiply by 9.
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.
9/9 is 1.
So just sin B is left.
So sin B = 9 × sin 50° over 12.
The calculator gives sin 50°: about 0.7660.
9 times 0.7660/12 is about 0.5745.
5 Find B
The sin⁻¹ button turns the ratio back into the angle.
The calculator gives about 35.067.
To the nearest tenth, B ≈ 35.1°.
6 Look for a second triangle
The calculator gives only one angle.
But 180° − 35.1° has the same sine: 144.9°.
It could be B too.
Does it fit?
With angle A, 50°, it makes 194.9°.
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 ≈ 35.1°.
7 Check
Let's check: put B back.
Both pairs must give the same number.
sin A over a is about 0.0638.
sin B over b: about 0.0639.
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.