Worked examples
The question
Question: find side b of the triangle with A = 40°, B = 60° and a = 8.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side a is across from angle A.
Side b, the one we want, is across from angle B.
2 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 = a over sin A.
bsin B = asin A
Careful!
Each side goes with the angle across from it, never with an angle next to it.
Careful: each side with the angle across from it
3 Put the numbers in
Now put the numbers in: B = 60°, a = 8 and A = 40°.
bsin 60° = 8sin 40°
4 Find b
b is on top, over sin 60°.
To remove the sin 60° at the bottom, we multiply by sin 60°.
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 60° × bsin 60° = sin 60° × 8sin 40°
sin 60° over sin 60° is 1.
So just b is left.
So b = 8 × sin 60° over sin 40°.
b = 8 × sin 60°sin 40°
The calculator gives sin 60°: about 0.8660.
And sin 40°: about 0.6428.
sin 60° ≈ 0.8660
sin 40° ≈ 0.6428
Put the numbers in.
b ≈ 8 × 0.86600.6428
That is about 10.778.
To the nearest tenth, b ≈ 10.8.
b ≈ 10.8
5 Check
Let's check: both pairs must give the same number.
b over sin 60° is about 12.47.
bsin 60° ≈ 10.80.8660 ≈ 12.47
a over sin 40° is about 12.45.
Almost the same: the gap is only from rounding.
asin 40° ≈ 80.6428 ≈ 12.45
And the bigger angle has the bigger side: 60° is more than 40°, and 10.8 is more than 8.
So it is right!
B > A, so b > a
Let's look back.
First: match sides and angles.
Then: write the law of sines.
Then: put the numbers in.
Then: find b.
Last: check.
▶ Watch it solved
The question
Question: find side a of the triangle with A = 50°, C = 70° and c = 12 cm.
1 Match sides and angles
Each side has a partner: the angle across from it.
Side c is across from angle C.
Side a, the one we want, is across from angle A.
2 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: a over sin A = c over sin C.
asin A = csin C
Careful!
Each side goes with the angle across from it, never with an angle next to it.
Careful: each side with the angle across from it
3 Put the numbers in
Now put the numbers in: A = 50°, c = 12 and C = 70°.
asin 50° = 12sin 70°
4 Find a
a is on top, over sin 50°.
To remove the sin 50° at the bottom, we multiply by sin 50°.
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 50° × asin 50° = sin 50° × 12sin 70°
sin 50° over sin 50° is 1.
So just a is left.
So a = 12 × sin 50° over sin 70°.
a = 12 × sin 50°sin 70°
The calculator gives sin 50°: about 0.7660.
And sin 70°: about 0.9397.
sin 50° ≈ 0.7660
sin 70° ≈ 0.9397
Put the numbers in.
a ≈ 12 × 0.76600.9397
That is about 9.782.
To the nearest tenth, a ≈ 9.8 cm.
a ≈ 9.8 cm
5 Check
Let's check: both pairs must give the same number.
a over sin 50° is about 12.79.
asin 50° ≈ 9.80.7660 ≈ 12.79
c over sin 70° is about 12.77.
Almost the same: the gap is only from rounding.
csin 70° ≈ 120.9397 ≈ 12.77
And the bigger angle has the bigger side: 50° is less than 70°, and 9.8 is less than 12.
So it is right!
A < C, so a < c
Let's look back.
First: match sides and angles.
Then: write the law of sines.
Then: put the numbers in.
Then: find a.
Last: check.
▶ Watch it solved
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.
sin Bb = sin Aa
3 Put the numbers in
Now put the numbers in: b = 8, A = 30° and a = 5.
sin B8 = sin 30°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 × sin B8 = 8 × sin 30°5
8/8 is 1.
So just sin B is left.
So sin B = 8 × sin 30° over 5.
sin B = 8 × sin 30°5
The calculator gives sin 30°: exactly 0.5.
sin 30° = 0.5
8 times 0.5/5 is 0.8.
sin B = 0.8
5 Find B
The sin⁻¹ button turns the ratio back into the angle.
B = sin−1(0.8)
The calculator gives about 53.13.
To the nearest tenth, B ≈ 53.1°.
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.
180° − 53.1° = 126.9°
Does it fit?
With angle A, 30°, it makes 156.9°.
That is less than 180°: there is room for angle C.
30° + 126.9° = 156.9°
Careful!
The calculator shows one angle, but here two triangles fit.
Side a can swing in or out and still reach the bottom line.
Careful: not only B ≈ 53.1° ✗
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°.
B ≈ 53.1° or 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 Aa = sin 30°5 ≈ 0.1000
sin B over b: about 0.1000 for both angles.
The same, or very close from rounding.
So it is right!
sin 53.1°8 ≈ 0.1000
sin 126.9°8 ≈ 0.1000
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.
Answer B ≈ 53.1° or 126.9°
▶ Watch it solved
