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Amy Ferguson Moncure

IB AA · Unit 2 · 5B Rules of Trigonometry

Cosine Rule

Use the cosine rule for SAS and for SSS triangles.

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Key Ideas Easy Medium Hard

c² = a² + b² − 2ab cos C

cos C = a² + b² − c²2ab

What to Remember

Find a side

c² = a² + b² − 2ab cos C

Angle C is between sides a and b. Side c is opposite that angle.

Find an angle

cos C = a² + b² − c²2ab

A negative cosine means the angle is obtuse.

When to choose it

Use the cosine rule for two sides and the included angle, or for all three sides.

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

Easy

Two sides and the included angle

Square the side you are finding.

Example
In triangle ABC, angle C = 60°, a = 8 cm, and b = 11 cm. Find side c.

c² = a² + b² − 2 a b cos C

c² = (8)² + (11)² − 2 (8) (11) cos 60°

c² = 97.0

c = 9.85 cm

Answer: 9.85 cm

1
In triangle ABC, angle C = 60°, a = 7 cm, and b = 9 cm. Find c.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (7)² + (9)² − 2 (7) (9) cos 60°

c² = 67.0

c = 8.19 cm

Answer: 8.19 cm

2
In triangle ABC, angle C = 48°, a = 5 cm, and b = 6 cm. Find c.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (5)² + (6)² − 2 (5) (6) cos 48°

c² = 20.9

c = 4.57 cm

Answer: 4.57 cm

3
In triangle ABC, angle C = 35°, a = 10 cm, and b = 13 cm. Find c.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (10)² + (13)² − 2 (10) (13) cos 35°

c² = 56.0

c = 7.48 cm

Answer: 7.48 cm

4
In triangle ABC, angle C = 120°, a = 4 cm, and b = 7 cm. Find c.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (4)² + (7)² − 2 (4) (7) cos 120°

c² = 93.0

c = 9.64 cm

Answer: 9.64 cm

5
In triangle ABC, angle C = 40°, a = 9 cm, and b = 9 cm. Find c.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (9)² + (9)² − 2 (9) (9) cos 40°

c² = 37.9

c = 6.16 cm

Answer: 6.16 cm

Medium

Three sides

The side you name in the numerator is opposite the angle you find.

Example
In triangle ABC, a = 7 cm, b = 8 cm, and c = 9 cm. Find angle C.

cos C = a² + b² − c²2 a b

cos C = (7)² + (8)² − (9)²2 (7) (8)

cos C = 0.286

C = 73.4°

Answer: 73.4°

1
In triangle ABC, a = 6 cm, b = 7 cm, and c = 8 cm. Find angle C.
Show Answer

cos C = a² + b² − c²2 a b

cos C = (6)² + (7)² − (8)²2 (6) (7)

cos C = 0.250

C = 75.5°

Answer: 75.5°

2
In triangle ABC, a = 5 cm, b = 7 cm, and c = 6 cm. Find angle B.
Show Answer

cos B = 5² + 6² − 7²2 (5) (6)

cos B = 0.200

B = 78.5°

Answer: 78.5°

3
In triangle ABC, a = 9 cm, b = 10 cm, and c = 12 cm. Find angle C.
Show Answer

cos C = a² + b² − c²2 a b

cos C = (9)² + (10)² − (12)²2 (9) (10)

cos C = 0.206

C = 78.1°

Answer: 78.1°

4
In triangle ABC, a = 4 cm, b = 5 cm, and c = 7 cm. Find angle C.
Show Answer

cos C = a² + b² − c²2 a b

cos C = (4)² + (5)² − (7)²2 (4) (5)

cos C = −0.200

C = 101.5°

Answer: 101.5°

5
In triangle ABC, a = 8 cm, b = 11 cm, and c = 14 cm. Find angle C.
Show Answer

cos C = a² + b² − c²2 a b

cos C = (8)² + (11)² − (14)²2 (8) (11)

cos C = −0.0625

C = 93.6°

Answer: 93.6°

Hard

An obtuse angle

Cosine is negative when the included angle is greater than 90°.

Example
In triangle ABC, angle C = 125°, a = 6 cm, and b = 10 cm. Find side c.

c² = a² + b² − 2 a b cos C

c² = (6)² + (10)² − 2 (6) (10) cos 125°

c² = 205

c = 14.3 cm

Answer: 14.3 cm

1
In triangle ABC, a = 5 cm, b = 6 cm, and c = 9 cm. Find angle C.
Show Answer

cos C = a² + b² − c²2 a b

cos C = (5)² + (6)² − (9)²2 (5) (6)

cos C = −0.333

C = 109.5°

Answer: 109.5°

2
In triangle PQR, p = 8 cm, q = 5 cm, and r = 7 cm. Find angle R.
Show Answer

cos R = 8² + 5² − 7²2 (8) (5)

cos R = 0.500

R = 60°

Answer: 60°

3
In triangle ABC, angle C = 50°, a = 7 cm, and b = 9 cm. Find side c to 3 significant figures.
Show Answer

c² = a² + b² − 2 a b cos C

c² = (7)² + (9)² − 2 (7) (9) cos 50°

c² = 49.0

c = 7.00 cm

Answer: 7.00 cm

4
In triangle ABC, angle A = 108°, b = 7 cm, and c = 12 cm. Find side a.
Show Answer

a² = b² + c² − 2 b c cos A

a² = (7)² + (12)² − 2 (7) (12) cos 108°

a² = 245

a = 15.6 cm

Answer: 15.6 cm

5
In triangle ABC, a = 13 cm, b = 14 cm, and c = 15 cm. Find angle A.
Show Answer

cos A = b² + c² − a²2 b c

cos A = (14)² + (15)² − (13)²2 (14) (15)

cos A = 0.600

A = 53.1°

Answer: 53.1°

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