Law of Cosines Calculator

Solve any triangle using the Law of Cosines: c² = a² + b² - 2ab·cos(C). Find all sides, angles, area, and perimeter.

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Triangle Solution
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complete solution below
Side a --
Side b --
Side c --
Angle A --
Angle B --
Angle C --
Area --
Perimeter --

Step-by-Step Solution

c² = a² + b² - 2ab·cos(C)

Understanding the Law of Cosines

The Law of Cosines (also called the Cosine Rule) is a fundamental formula in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem, which is a special case when the angle C equals 90 degrees.

The law is used to solve triangles when you know either all three sides (SSS) or two sides and the included angle (SAS). It is one of the most important tools in surveying, navigation, physics, and engineering.

Law of Cosines Formulas

Finding Side c

Given sides a, b and the included angle C.

c² = a² + b² - 2ab·cos(C)

Finding Side a

Given sides b, c and the included angle A.

a² = b² + c² - 2bc·cos(A)

Finding Side b

Given sides a, c and the included angle B.

b² = a² + c² - 2ac·cos(B)

Finding Angle C

Given all three sides a, b, c (SSS case).

cos(C) = (a² + b² - c²) / (2ab)

Finding Angle A

Given all three sides a, b, c (SSS case).

cos(A) = (b² + c² - a²) / (2bc)

Finding Angle B

Given all three sides a, b, c (SSS case).

cos(B) = (a² + c² - b²) / (2ac)

When to Use the Law of Cosines

  • SSS (Side-Side-Side): When you know all three sides of a triangle and need to find the angles.
  • SAS (Side-Angle-Side): When you know two sides and the angle between them, and need to find the third side and remaining angles.

For other cases like ASA (Angle-Side-Angle) or AAS (Angle-Angle-Side), the Law of Sines is typically more convenient to use first.

Relationship to the Pythagorean Theorem

When angle C = 90 degrees, cos(C) = cos(90°) = 0, and the Law of Cosines reduces to c² = a² + b², which is exactly the Pythagorean theorem. This shows that the Pythagorean theorem is a special case of the more general Law of Cosines.

Practical Applications

  • Surveying: Calculating distances between points when direct measurement is impossible.
  • Navigation: Determining the distance between two locations given bearings and a known baseline.
  • Physics: Resolving forces in vector addition, computing resultant velocities.
  • Construction: Verifying angles and distances in building frames and roof trusses.
  • Astronomy: Calculating distances between celestial objects using triangulation methods.

Triangle Inequality

For three lengths to form a valid triangle, each side must be less than the sum of the other two sides: a + b > c, a + c > b, and b + c > a. If this condition is not met, no triangle can be formed.

Tips for Using the Law of Cosines

  • Always check the triangle inequality before computing.
  • When finding angles from SSS, find the largest angle first (opposite the longest side) to avoid ambiguity.
  • The sum of all angles in a triangle must equal 180°. Use this as a check.
  • For obtuse triangles, one angle will be greater than 90°, and its cosine will be negative.
  • Use Heron's formula for area: Area = √(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2.