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Cosine Rule (Law of Cosines)
Trigonometry • Grades 9, 10, 11
Trigonometry
Cosine Rule (Law of Cosines)
Solve triangles with 3 sides or 2 sides + included angle.
The Formula
c² = a² + b² - 2ab·cos(C)
Used for: Finding an unknown side (SAS) or angle (SSS) in any general oblique triangle
a, b, cLengths of triangle sides
CAngle opposite to side c (in degrees or radians)
⚡Live Interactive Quick Solver
Instant CalculationOpposite Side (c):3.61 units
Step: c² = 3² + 4² - 2(3)(4)cos(60°) = 13.00 ➔ c = 3.61
c^2 = a^2 + b^2 - 2ab \cos(C)
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🎯 When to Use the Cosine Rule
Best for these two scenarios:
📐
Works for ANY triangleNot just right triangles!
📊
Two variationsFind a side OR find an angle
🎯
Need 3 pieces of infoEither 3 sides or 2 sides + included angle
📝 Worked Examples
Find a Side (SAS)
Given:
a = 8, b = 6, C = 60°. Find c.1
Formula
c² = a² + b² - 2ab·cos(C)2
Substitute
c² = 64 + 36 - 2(8)(6)·cos(60°)3
Calculate
c² = 100 - 96(0.5) = 524
Square root
c = √52 ≈ 7.21Answer: c ≈ 7.21
Find an Angle (SSS)
Given:
a = 5, b = 7, c = 8. Find angle C.1
Rearrange
cos(C) = (a² + b² - c²) / 2ab2
Substitute
cos(C) = (25 + 49 - 64) / 703
Calculate
cos(C) = 10/70 = 0.1434
Inverse cos
C = cos⁻¹(0.143) ≈ 81.8°Answer: C ≈ 81.8°
💡 Pro Tips
Special case: C = 90°
When C = 90°, cos(C) = 0, and this becomes the Pythagorean theorem!
SAS vs SSS
SAS (2 sides + angle) → find missing side. SSS (3 sides) → find any angle.
vs Sine Rule
Use cosine rule when you don't have matching angle-side pairs
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