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The cosine rule practice questions with answers

Higher · Grade 7 · Calculator

A second set of the cosine rule questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

Use the cosine rule with included angle 60 degrees, enclosing sides 8 and 8. What is the opposite side?

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Answer: 8

  1. For these equal sides, cos 60 = 1/2.
  2. The opposite side equals 8.
  • Method (1): For these equal sides, cos 60 = 1/2.
  • Accuracy (1): Answer 8

Question 2 Easy

In triangle ABC, the sides are 7, 8 and 9. Find the angle opposite 9 to the nearest degree.

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Answer: 73

  1. cos C = (7² + 8² − 9²) / (2×7×8).
  2. The angle is 73 degrees.
  • Method (1): cos C = (7² + 8² − 9²) / (2×7×8).
  • Accuracy (1): Answer 73

Question 3 Medium

Use the cosine rule with included angle 90 degrees, enclosing sides 6 and 8. What is the opposite side?

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Answer: 10

  1. cos 90 = 0, so the rule is Pythagoras here.
  2. The opposite side is 10.
  • Method (1): cos 90 = 0, so the rule is Pythagoras here.
  • Accuracy (1): Answer 10

Question 4 Medium

A ship sails 8 km due north from A to B, then 6 km on a bearing of 70 degrees from B to C. Find AC to 3 significant figures.

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Answer: 11.5

  1. The angle at B inside the triangle is 110 degrees.
  2. Use the cosine rule on 8, 6 and 110 degrees.
  • Method (1): The angle at B inside the triangle is 110 degrees.
  • Accuracy (1): Answer 11.5

Question 5 Hard

In triangle ABC, the sides are 2, 3 and 4. Find the angle opposite 4 to the nearest degree.

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Answer: 104

  1. cos C = (2² + 3² − 4²) / (2×2×3) = −0.25.
  2. The angle is 104 degrees.
  • Method (1): cos C = (2² + 3² − 4²) / (2×2×3) = −0.25.
  • Accuracy (1): Answer 104

Question 6 Hard

A ship sails 10 km due north from A to B, then 9 km on a bearing of 100 degrees from B to C. Find AC to 3 significant figures.

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Answer: 12.2

  1. The included angle at B is 80 degrees.
  2. The cosine rule gives AC.
  • Method (1): The included angle at B is 80 degrees.
  • Accuracy (1): Answer 12.2

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 13 marks

In triangle ABC, sides 4 and 6 enclose an angle of 60 degrees. Find the opposite side to 3 significant figures.

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Answer: 5.29

  1. c² = 4² + 6² − 2×4×6×cos 60.
  2. Find the square root.
  • Method (1): c² = 4² + 6² − 2×4×6×cos 60.
  • Method (1): Find the square root.
  • Accuracy (1): Answer 5.29

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What is the exact value of cos 120°?

  1. -1/2
  2. 1/2
  3. -√3/2
  4. 0
Show answer

-1/2 cos 120° = -cos 60° = -1/2.

Check 2

Two sides of length 7 enclose an angle of 120°. What is the opposite side length?

  1. 7√3
  2. 7
  3. 14
  4. 7√2
Show answer

7√3 c² = 49 + 49 - 98(-1/2) = 147, so c = 7√3.

Check 3

Why does an included angle of 120° make the opposite side longer than an acute angle?

  1. cos 120° is negative so -2ab cos C becomes positive
  2. cos 120° is zero so 2ab drops out
  3. 120° makes the sides shorter
  4. Pythagoras squares the hypotenuse
Show answer

cos 120° is negative so -2ab cos C becomes positive The -2ab cos C term is added when cos C is negative.

Check 4

Two sides of length 5 enclose an angle of 60°. What is the opposite side length?

  1. 5
  2. 5√3
  3. 10
  4. 25
Show answer

5 An isosceles triangle with a 60° vertex is equilateral, so the opposite side is 5.

Check 5

What is the cosine of the angle opposite side 4 in a triangle with sides 2, 3 and 4?

  1. -1/4
  2. 1/4
  3. -3/8
  4. -1/2
Show answer

-1/4 cos C = (4 + 9 - 16) / 12 = -3/12 = -1/4.

Check 6

In a triangle with sides 2, 3 and 4, is the angle opposite side 4 acute or obtuse?

  1. obtuse
  2. acute
  3. right-angled
  4. reflex
Show answer

obtuse The cosine is negative (-1/4), so the angle is obtuse.

Check 7

When does the cosine rule c² = a² + b² − 2ab cos C reduce to Pythagoras’ theorem?

  1. when C = 90° and cos 90° = 0
  2. when C = 60° and cos 60° = 1/2
  3. when C = 120° and cos 120° = -1/2
  4. when C = 0° and cos 0° = 1
Show answer

when C = 90° and cos 90° = 0 At 90°, cos 90° = 0 so the -2ab cos C term vanishes.

Check 8

Two sides of length 4 enclose an angle of 120°. What is the opposite side length?

  1. 4√3
  2. 4
  3. 8
  4. 4√2
Show answer

4√3 c² = 16 + 16 + 16 = 48, so c = √48 = 4√3.

Check 9

A student uses cos 120° = +1/2 in the cosine rule. What mistake did they make?

  1. forgot that cosine is negative for obtuse angles
  2. forgot to square the sides
  3. multiplied by 2 twice
  4. used sine instead of cosine
Show answer

forgot that cosine is negative for obtuse angles cos 120° is in the second quadrant where cosine is negative (-1/2).

Check 10

Two sides of length 9 enclose an angle of 60°. What is the third side length?

  1. 9
  2. 9√3
  3. 18
  4. 9√2
Show answer

9 Equal sides and 60° form an equilateral triangle with third side 9.

Check 11

Write the cosine rule formula rearranged to find cos C.

  1. cos C = (a² + b² − c²) / (2ab)
  2. cos C = (a² + b² + c²) / (2ab)
  3. cos C = (c² − a² − b²) / (2ab)
  4. cos C = 2ab / (a² + b² − c²)
Show answer

cos C = (a² + b² − c²) / (2ab) c² = a² + b² - 2ab cos C rearranges to 2ab cos C = a² + b² - c².

Check 12

Two sides of length 2 enclose an angle of 120°. What is the opposite side length?

  1. 2√3
  2. 2
  3. 4
  4. 2√2
Show answer

2√3 c² = 4 + 4 + 4 = 12, so c = √12 = 2√3.

Check 13

Why does 4² > 2² + 3² show that the angle opposite side 4 is obtuse?

  1. c² > a² + b² makes the numerator of cos C negative
  2. c² > a² + b² makes the denominator negative
  3. it proves the triangle is impossible
  4. it proves all angles are acute
Show answer

c² > a² + b² makes the numerator of cos C negative cos C = (a² + b² - c²)/(2ab) has numerator 13 - 16 = -3 < 0.

Check 14

Two sides of length 10 enclose an angle of 120°. What is the opposite side length?

  1. 10√3
  2. 10
  3. 20
  4. 10√2
Show answer

10√3 c² = 100 + 100 + 100 = 300, so c = 10√3.

Check 15

Which angle is used in the cosine rule c² = a² + b² − 2ab cos C?

  1. the included angle between sides a and b
  2. the angle opposite side a
  3. the angle opposite side b
  4. any angle in the triangle
Show answer

the included angle between sides a and b Angle C is the angle enclosed between the known sides a and b.

Keep going with The cosine rule

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