Trigonometry

Law of Cosines

Formula

c² = a² + b² − 2ab × cos(C)

What is Law of Cosines?

The Law of Cosines relates the sides of any triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem to non-right triangles.

Variables

a
a — Side opposite angle A
b
b — Side opposite angle B
c
c — Side opposite angle C
C
C — Angle between sides a and b

Worked Examples

Example 1

In triangle ABC, a = 7, b = 9, and C = 60°. Find side c.

Solution:

c² = 49 + 81 − 2(7)(9)(cos 60°) = 130 − 126(0.5) = 130 − 63 = 67. c = √67 ≈ 8.19

Frequently Asked Questions

Related Formulas

Trigonometric Ratios (sin, cos, tan)

sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent

a/sin(A) = b/sin(B) = c/sin(C)

Pythagorean Trigonometric Identities

sin²θ + cos²θ = 1

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