Law of Cosines Calculator — Solve Any Triangle
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How the Law of Cosines Works
The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of any triangle to the cosine of one of its angles. It generalizes the Pythagorean theorem to non-right triangles and is attributed to the ancient Greek mathematician Euclid, who presented an equivalent geometric proposition in his Elements (Book II, Proposition 12-13) around 300 BCE. The modern algebraic form was developed by the French mathematician Francois Viete in the 16th century.
According to Wolfram MathWorld, the Law of Cosines is essential whenever you need to solve a triangle from SAS (two sides and the included angle) or SSS (all three sides) information. It is widely used in navigation, surveying, astronomy, structural engineering, and computer graphics. GPS systems use triangulation principles derived from this law to calculate positions. The companion theorem, the Law of Sines, handles different triangle configurations (ASA, AAS). Together, these two laws can solve any triangle given sufficient information. For right triangles specifically, use our Right Triangle Calculator.
The Law of Cosines Formula
The Law of Cosines has three equivalent forms, one for each side of the triangle:
- c^2 = a^2 + b^2 - 2ab cos(C) -- finds side c when you know sides a, b and included angle C
- b^2 = a^2 + c^2 - 2ac cos(B) -- finds side b when you know sides a, c and included angle B
- a^2 = b^2 + c^2 - 2bc cos(A) -- finds side a when you know sides b, c and included angle A
To find an angle when all three sides are known, rearrange: cos(C) = (a^2 + b^2 - c^2) / (2ab)
Worked example (SAS): Given a = 5, b = 7, C = 60 degrees. c^2 = 25 + 49 - 2(5)(7)cos(60) = 74 - 70(0.5) = 74 - 35 = 39. So c = sqrt(39) = 6.245 units. Area = 0.5 x 5 x 7 x sin(60) = 15.155 sq units.
Key Terms You Should Know
- SAS (Side-Angle-Side) -- knowing two sides and the included angle (the angle between them). This is the primary use case for the Law of Cosines.
- SSS (Side-Side-Side) -- knowing all three sides. Rearrange the formula to find each angle using the inverse cosine function.
- Included angle -- the angle formed between two known sides. The Law of Cosines requires the included angle, not any arbitrary angle.
- Obtuse angle -- an angle greater than 90 degrees. When C > 90, cos(C) is negative, making c^2 larger than a^2 + b^2 (the side opposite an obtuse angle is the longest).
- Triangle inequality -- for a valid triangle, the sum of any two sides must exceed the third side. The Law of Cosines automatically satisfies this for valid inputs.
Law of Cosines vs. Law of Sines
Choosing between these two fundamental triangle-solving laws depends on what information you have. This table summarizes when to use each.
| Known Information | Use Which Law | Notes |
|---|---|---|
| SAS (2 sides + included angle) | Law of Cosines | Always gives 1 unique solution |
| SSS (3 sides) | Law of Cosines | Rearrange to find angles |
| ASA (2 angles + included side) | Law of Sines | Find 3rd angle first (180 - A - B) |
| AAS (2 angles + non-included side) | Law of Sines | Find 3rd angle first |
| SSA (2 sides + non-included angle) | Law of Sines (caution) | Ambiguous case: 0, 1, or 2 solutions |
Practical Law of Cosines Examples
Surveying a lake: A surveyor stands at point A and measures distances to two points B and C on the opposite shore: AB = 200 m and AC = 150 m. The angle BAC = 75 degrees. Using the Law of Cosines: BC^2 = 200^2 + 150^2 - 2(200)(150)cos(75) = 40,000 + 22,500 - 60,000(0.2588) = 47,002. BC = 216.8 m. The distance across the lake is approximately 217 meters.
Finding an angle (SSS): A triangular lot has sides 30 ft, 40 ft, and 45 ft. To find the largest angle (opposite the longest side): cos(C) = (30^2 + 40^2 - 45^2) / (2 x 30 x 40) = (900 + 1600 - 2025) / 2400 = 475/2400 = 0.1979. C = arccos(0.1979) = 78.6 degrees. Calculate the area using the Triangle Area Calculator.
Navigation: A ship sails 12 km on a bearing of 040 degrees, then 8 km on a bearing of 120 degrees. The angle between the two legs is 120 - 40 = 80 degrees. Distance from start: d^2 = 12^2 + 8^2 - 2(12)(8)cos(80) = 144 + 64 - 192(0.1736) = 174.67. d = 13.22 km from the starting point.
Tips for Solving Law of Cosines Problems
- Always identify your configuration first: Check if you have SAS or SSS. If you have AAS, ASA, or SSA, use the Law of Sines instead.
- Convert degrees to radians if needed: Most scientific calculators and programming languages use radians. Multiply degrees by pi/180 to convert. 60 degrees = pi/3 radians.
- For SSS, find the largest angle first: The largest angle is always opposite the longest side. This avoids potential issues with inverse cosine returning only acute angles.
- Verify with angle sum: After finding all three angles, confirm they sum to exactly 180 degrees. If not, check for rounding errors in intermediate steps.
- Remember the Pythagorean connection: When C = 90 degrees, cos(90) = 0, and the formula simplifies to c^2 = a^2 + b^2. Use our Pythagorean Theorem Calculator for right triangle cases.
Frequently Asked Questions
When should I use the Law of Cosines instead of the Law of Sines?
Use the Law of Cosines when you know SAS (two sides and the included angle between them) or SSS (all three side lengths). These are the two configurations where the Law of Sines cannot be directly applied. For example, if you know sides a = 8, b = 12, and the angle C = 50 degrees between them, the Law of Cosines gives you side c directly. Use the Law of Sines when you know ASA, AAS, or SSA -- but beware the ambiguous case with SSA, which can produce zero, one, or two valid triangles.
Can the Law of Cosines produce negative results?
The final result (a side length) is always positive for valid triangles because you take the square root of c^2. However, the intermediate expression inside the square root can involve a negative cosine term. When angle C is obtuse (greater than 90 degrees), cos(C) is negative, making the -2ab cos(C) term positive and increasing c^2. For example, if C = 120 degrees, cos(120) = -0.5, so -2ab(-0.5) = +ab, giving c^2 = a^2 + b^2 + ab. This correctly produces a longer side opposite the obtuse angle.
How is the Law of Cosines related to the Pythagorean theorem?
The Pythagorean theorem is a special case of the Law of Cosines where angle C = 90 degrees. Since cos(90) = 0, the term -2ab cos(C) vanishes, reducing c^2 = a^2 + b^2 - 2ab cos(90) to simply c^2 = a^2 + b^2. This makes the Law of Cosines the more general principle, applicable to all triangles -- acute, right, and obtuse. The Pythagorean theorem was known to ancient Babylonians around 1800 BCE, while the general cosine law was first explicitly stated by Euclid around 300 BCE.
How do I find all three angles using the Law of Cosines?
When you know all three sides (SSS), rearrange the formula to solve for each angle: cos(A) = (b^2 + c^2 - a^2) / (2bc), then A = arccos(result). Repeat for angle B using cos(B) = (a^2 + c^2 - b^2) / (2ac). Find the third angle as C = 180 - A - B. For example, with sides 5, 7, and 8: cos(A) = (49 + 64 - 25) / (2 x 7 x 8) = 88/112 = 0.786, so A = 38.2 degrees. This method always produces unique, unambiguous results for valid triangles.
What real-world applications use the Law of Cosines?
The Law of Cosines is used extensively in professional fields. Land surveyors use it to calculate distances across obstacles (lakes, buildings) from two measured sides and an angle. Navigators use it for dead reckoning -- determining position after sailing or flying two legs at known angles. Astronomers apply it in the celestial triangle to compute distances between stars. Structural engineers use it to analyze force triangles in trusses and bridges. GPS receivers use related triangulation mathematics to compute your position from satellite signals. Even video game developers use it for collision detection and character pathfinding.