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Law of Cosines

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

The Law of Cosines, c^2 = a^2 + b^2 - 2ab*cosC, relates the three sides of any triangle to one of its angles. Use it in the SAS case (two sides and the included angle) to find the third side, and rearrange it to cosC = (a^2 + b^2 - c^2)/(2ab) in the SSS case (three sides) to find an angle. The angle C always sits opposite the side c, and when C = 90 degrees the formula collapses to the Pythagorean theorem.

What you'll learn

Why it matters: Surveyors and GPS systems use the Law of Cosines to compute a distance across terrain they cannot walk directly, when only two leg distances and the angle between them are known.

Worked example

Problem. In triangle ABC, a = 8, b = 11, and the included angle C = 37 degrees. Find side c to the nearest tenth.

  1. Apply c^2 = a^2 + b^2 - 2ab*cosC = 8^2 + 11^2 - 2(8)(11)cos(37)
  2. Compute: 64 + 121 - 176(0.7986) = 185 - 140.6 = 44.4
  3. Take the square root: c = sqrt(44.4) = 6.7

Answer: 6.7

Practice problems

1. In triangle ABC, a = 5, b = 7, and the included angle C = 90 degrees. Find side c to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In triangle ABC, a = 5, b = 7, and the included angle C = 90 degrees. Find side c to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. c^2 = 5^2 + 7^2 - 2(5)(7)cos(90) = 25 + 49 - 70(0)
  4. Since cos(90) = 0, c^2 = 74
  5. c = sqrt(74) = 8.6
  6. Check the result by substituting or estimating: the response should match 8.6 and make sense in the original problem.

Answer: 8.6

2. You know two sides of a triangle and the angle between them and want the third side. Which equation should you use?

Choices: c^2 = a^2 + b^2 - 2ab*cosC · c^2 = a^2 + b^2 · cosC = (a^2 + b^2 - c^2)/(2ab) · c = a + b - 2ab

Show solution
  1. Warm-up: First identify exactly what the question is asking: You know two sides of a triangle and the angle between them and want the third side. Which equation should you use?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Two sides plus the included angle is the SAS case for finding a side.
  4. The Law of Cosines c^2 = a^2 + b^2 - 2ab*cosC directly gives the third side.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: c^2 = a^2 + b^2 - 2ab*cosC

3. In triangle ABC, a = 6, b = 10, and the included angle C = 60 degrees. Find side c to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: In triangle ABC, a = 6, b = 10, and the included angle C = 60 degrees. Find side c to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. c^2 = 6^2 + 10^2 - 2(6)(10)cos(60) = 36 + 100 - 120(0.5)
  4. c^2 = 136 - 60 = 76
  5. c = sqrt(76) = 8.7
  6. Check the result by substituting or estimating: the response should match 8.7 and make sense in the original problem.

Answer: 8.7

4. In triangle ABC, a = 9, b = 12, and the included angle C = 45 degrees. Find side c to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: In triangle ABC, a = 9, b = 12, and the included angle C = 45 degrees. Find side c to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. c^2 = 9^2 + 12^2 - 2(9)(12)cos(45) = 81 + 144 - 216(0.7071)
  4. c^2 = 225 - 152.7 = 72.3
  5. c = sqrt(72.3) = 8.5
  6. Check the result by substituting or estimating: the response should match 8.5 and make sense in the original problem.

Answer: 8.5

5. In triangle ABC, a = 14, b = 10, and the included angle C = 120 degrees. Find side c to the nearest tenth.

Show solution
  1. Core Practice: First identify exactly what the question is asking: In triangle ABC, a = 14, b = 10, and the included angle C = 120 degrees. Find side c to the nearest tenth.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. c^2 = 14^2 + 10^2 - 2(14)(10)cos(120) = 196 + 100 - 280(-0.5)
  4. c^2 = 296 + 140 = 436
  5. c = sqrt(436) = 20.9
  6. Check the result by substituting or estimating: the response should match 20.9 and make sense in the original problem.

Answer: 20.9

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