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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.

Applications ask students to choose the right trig model: Law of Sines, Law of Cosines, SSA analysis, triangle area, vectors, periodic modeling, or a cumulative mix of earlier tools.

What you'll learn

Why it matters: Surveyors, pilots, engineers, architects, and data modelers use trig when right triangles are not enough.

Worked example

Problem. Law of Cosines: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.

  1. Use Law of Cosines.
  2. 100 = 36 + 64 - 96cos(C).
  3. cos(C)=0, so C=90 degrees.

Answer: 90

Practice problems

1. Law of Cosines: In triangle ABC, A = 40 degrees, B = 65 degrees, and a = 12. Find C.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Law of Cosines: In triangle ABC, A = 40 degrees, B = 65 degrees, and a = 12. Find C.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Triangle angles sum to 180 degrees.
  4. C = 180 - 40 - 65.
  5. C = 75 degrees.
  6. Check the result by substituting or estimating: the response should match 75 and make sense in the original problem.

Answer: 75

2. Law of Cosines: Use Law of Sines: A = 30 degrees, B = 45 degrees, a = 10. Find b to the nearest tenth.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Law of Cosines: Use Law of Sines: A = 30 degrees, B = 45 degrees, a = 10. Find b to the nearest tenth.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set b/sin(45) = 10/sin(30).
  4. b = 10sin(45)/sin(30).
  5. b is about 14.1.
  6. Check the result by substituting or estimating: the response should match 14.1 and make sense in the original problem.

Answer: 14.1

3. Law of Cosines: Use Law of Cosines with a = 5, b = 8, and C = 60 degrees. Find c.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Law of Cosines: Use Law of Cosines with a = 5, b = 8, and C = 60 degrees. Find c.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. c^2 = 25 + 64 - 2(5)(8)(1/2).
  4. c^2 = 49.
  5. c = 7.
  6. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

4. Law of Cosines: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Law of Cosines: A triangle has sides 6, 8, and 10. Find the angle opposite side 10.
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use Law of Cosines.
  4. 100 = 36 + 64 - 96cos(C).
  5. cos(C)=0, so C=90 degrees.
  6. Check the result by substituting or estimating: the response should match 90 and make sense in the original problem.

Answer: 90

5. Law of Cosines: In SSA, why can two triangles be possible?

Choices: sine has the same positive value for an acute and an obtuse angle · cosine is never negative · all sides are equal · angles can sum over 180 degrees

Show solution
  1. Core Practice: First identify exactly what the question is asking: Law of Cosines: In SSA, why can two triangles be possible?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The Law of Sines may give sin(B).
  4. Both B and 180-B can share that sine value.
  5. Check the triangle sum.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: sine has the same positive value for an acute and an obtuse angle

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