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Amy Ferguson Moncure

IB AA · Unit 2 · 5B Rules of Trigonometry

Sine Rule

Use the sine rule when you know a side and the angle opposite it.

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Key Ideas Easy Medium Hard

asin A = bsin B

What to Remember

Sine rule

asin A = bsin B = csin C

Side a is opposite angle A. The same letter names the side and the angle across from it.

Find a side

You need the angle opposite the known side and the angle opposite the side you want.

If you know two angles, the third is 180° minus their sum.

Find an angle

sin B = b × sin Aa

These problems ask for the acute angle.

Easy

Find a side

Pair each side with the sine of the angle opposite it.

Example
In triangle ABC, angle A = 42°, angle B = 68°, and side a = 11 cm. Find side b.

asin A = bsin B

11sin 42° = bsin 68°

b = 11 × sin 68°sin 42°

b = 15.2 cm

Answer: 15.2 cm

1
In triangle ABC, angle A = 50°, angle B = 60°, and a = 9 cm. Find b.
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asin A = bsin B

9sin 50° = bsin 60°

b = 9 × sin 60°sin 50°

b = 10.2 cm

Answer: 10.2 cm

2
In triangle ABC, angle A = 35°, angle B = 80°, and a = 14 cm. Find b.
Show Answer

asin A = bsin B

14sin 35° = bsin 80°

b = 14 × sin 80°sin 35°

b = 24.0 cm

Answer: 24.0 cm

3
In triangle ABC, angle A = 47°, angle B = 58°, and a = 20 cm. Find b.
Show Answer

asin A = bsin B

20sin 47° = bsin 58°

b = 20 × sin 58°sin 47°

b = 23.2 cm

Answer: 23.2 cm

4
In triangle ABC, angle A = 25°, angle B = 100°, and a = 8 cm. Find b.
Show Answer

asin A = bsin B

8sin 25° = bsin 100°

b = 8 × sin 100°sin 25°

b = 18.6 cm

Answer: 18.6 cm

5
In triangle ABC, angle A = 53°, angle B = 64°, and a = 16 cm. Find b.
Show Answer

asin A = bsin B

16sin 53° = bsin 64°

b = 16 × sin 64°sin 53°

b = 18.0 cm

Answer: 18.0 cm

Medium

Find an acute angle

Rearrange so sin of the unknown angle is the result.

Example
In triangle ABC, angle A = 55°, a = 12 cm, and b = 9 cm. Find acute angle B.

sin Bb = sin Aa

sin B = 9 × sin 55°12

sin B = 0.614

B = 37.9°

Answer: 37.9°

1
In triangle PQR, angle P = 48°, p = 15 cm, and q = 11 cm. Find acute angle Q.
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sin Qb = sin Aa

sin Q = 11 × sin 48°15

sin Q = 0.545

B = 33.0°

Answer: 33.0°

2
In triangle PQR, angle P = 62°, p = 18 cm, and q = 13 cm. Find acute angle Q.
Show Answer

sin Qb = sin Aa

sin Q = 13 × sin 62°18

sin Q = 0.638

B = 39.6°

Answer: 39.6°

3
In triangle PQR, angle P = 37°, p = 10 cm, and q = 7.50 cm. Find acute angle Q.
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sin Qb = sin Aa

sin Q = 7.50 × sin 37°10

sin Q = 0.451

B = 26.8°

Answer: 26.8°

4
In triangle PQR, angle P = 71°, p = 22 cm, and q = 16 cm. Find acute angle Q.
Show Answer

sin Qb = sin Aa

sin Q = 16 × sin 71°22

sin Q = 0.688

B = 43.4°

Answer: 43.4°

5
In triangle PQR, angle P = 44°, p = 9 cm, and q = 6 cm. Find acute angle Q.
Show Answer

sin Qb = sin Aa

sin Q = 6 × sin 44°9

sin Q = 0.463

B = 27.6°

Answer: 27.6°

Hard

Find the third angle first

The angles of a triangle add to 180°.

Example
In triangle ABC, angle A = 31°, angle B = 112°, and side a = 7.2 cm. Find side b.

Angle C = 180° − 31° − 112°

Angle C = 37°

asin A = bsin B

b = 7.2 × sin 112°sin 31°

b = 13.0 cm

Answer: 13.0 cm

1
In triangle ABC, angle A = 28°, angle B = 105°, and a = 9.40 cm. Find b.
Show Answer

9.40sin 28° = bsin 105°

b = 9.40 × sin 105°sin 28°

b = 19.3 cm

Answer: 19.3 cm

2
In triangle ABC, angle A = 41°, angle B = 97°, and a = 13 cm. Find b.
Show Answer

13sin 41° = bsin 97°

b = 13 × sin 97°sin 41°

b = 19.7 cm

Answer: 19.7 cm

3
In triangle ABC, angle A = 36°, angle B = 77°, and a = 10 cm. Find side c.
Show Answer

C = 180° − 36° − 77°

C = 67°

c = 10 × sin 67°sin 36°

c = 15.7 cm

Answer: 15.7 cm

4
In triangle ABC, angle A = 22°, angle B = 118°, and a = 6.50 cm. Find side c.
Show Answer

C = 180° − 22° − 118°

C = 40°

c = 6.50 × sin 40°sin 22°

c = 11.2 cm

Answer: 11.2 cm

5
In triangle ABC, angle A = 49°, angle C = 63°, and side b = 18 cm. Find side a.
Show Answer

B = 180° − 49° − 63°

B = 68°

a = 18 × sin 49°sin 68°

a = 14.7 cm

Answer: 14.7 cm

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