Law of Cosines

Generalizing the Pythagorean Theorem to All Triangles

The Law of Cosines extends the Pythagorean theorem beyond right triangles, relating the lengths of all three sides of any triangle to one of its angles. This fundamental identity connects geometry, trigonometry, and spatial reasoning, enabling us to solve triangles and understand spatial relationships in the physical world.

Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos(C)

For any triangle with sides a, b, c and angle C opposite side c, this identity relates all three sides and one angle. When C = 90°, it reduces to the Pythagorean theorem.

$a, b, c$

The lengths of the three sides of the triangle.

$C$

The angle opposite side c, measured in radians or degrees.

$\cos(C)$

The cosine of angle C, which adjusts the formula based on the angle.

$2ab$

Twice the product of sides a and b, the correction term.

Prerequisites & Learning Path

Before diving deep into Law of Cosines, it's helpful to have a solid understanding of foundational concepts. Here are the key prerequisites that will help you master this identity.

Basic Geometry

Understanding of points, lines, angles, and basic shapes.

Required

Coordinate Geometry

Understanding of the coordinate plane and graphing.

Recommended

Similarity and Congruence

Understanding of geometric transformations.

Recommended

Historical Timeline

6th century BCE

Pythagoras and his school develop fundamental geometric theorems.

3rd century BCE

Euclid writes the Elements, systematizing geometry.

17th century

Descartes introduces coordinate geometry, connecting algebra and geometry.

19th century

Non-Euclidean geometries are discovered, revolutionizing mathematical thought.

The complete history of the Law of Cosines

The Law of Cosines has ancient origins, appearing in various forms across different mathematical traditions. The earliest known geometric statement appears in Euclid's Elements (c. 300 BCE) as Propositions II.12 and II.13, which state relationships for obtuse and acute triangles, though not in the modern trigonometric form we use today.

The trigonometric version was developed during the Islamic Golden Age (8th–14th centuries). Persian mathematician Al-Khwarizmi (c. 780–850 CE) worked on spherical trigonometry, and later mathematicians like Jamshīd al-Kāshī (c. 1380–1429) developed comprehensive trigonometric tables and relationships. The law was essential for astronomical calculations and navigation.

In 15th-century Europe, mathematicians like Regiomontanus (1436–1476) and François Viète (1540–1603) further developed the law. Viète, in particular, contributed to the modern notation and comprehensive treatment. The law became fundamental to the development of trigonometry as a separate mathematical discipline in the 16th and 17th centuries.

The modern form using cosine was standardized in the 18th century with the development of analytic trigonometry. Leonhard Euler (1707–1783) contributed significantly to trigonometric identities, and the Law of Cosines became a cornerstone of triangle solving, essential for surveying, navigation, and engineering.

The law's connection to the Pythagorean theorem was recognized early: when angle C equals 90°, $\cos(90°) = 0$ , and the law reduces to $c^2 = a^2 + b^2$ , the familiar Pythagorean theorem. This connection reveals the Law of Cosines as a natural generalization of one of mathematics' most fundamental relationships.

What each symbol means: a deep dive

$a, b, c$

The three side lengths of the triangle. In the standard form, c is the side opposite angle C. The law can be written for any side-angle pair by relabeling the triangle. These represent physical distances in space, whether measured in meters, miles, or any unit of length. In applications, they might represent forces, vectors, or spatial relationships in physics and engineering.

$C$

The angle opposite side c, measured in radians or degrees. When $C = 90°$ , $\cos(90°) = 0$ , and the law reduces to the Pythagorean theorem: $c^2 = a^2 + b^2$ . When $C < 90°$ (acute), the cosine is positive, and when $C > 90°$ (obtuse), the cosine is negative, reflecting how the angle affects the relationship between sides.

$\cos(C)$

The cosine of angle C, which adjusts the formula based on the angle. Cosine measures the horizontal projection of a unit vector at angle C. In the Law of Cosines, it acts as a correction factor: when the angle is acute, it reduces the square of side c; when obtuse, it increases it. This reflects the geometric fact that obtuse angles "stretch" the opposite side.

$2ab$

Twice the product of sides a and b. This term represents the interaction between the two sides adjacent to angle C. The factor of 2 comes from the geometric construction: when we drop a perpendicular and use coordinate geometry, the cross-term appears with coefficient 2. This term is what distinguishes the Law of Cosines from the simpler Pythagorean theorem.

Complete proof: step by step

Why use coordinate geometry? (A beginner's guide)

If you're new to advanced geometry, you might wonder: why are we using coordinate geometry instead of pure geometric reasoning? Here's the simple answer:

Coordinate geometry translates shapes into algebra. By placing a triangle in the coordinate plane, we can use the distance formula (which is just the Pythagorean theorem in disguise) to calculate side lengths. This approach is powerful because:

We can use algebra to manipulate geometric relationships
We can prove general cases that work for all triangles, not just special ones
The method connects geometry to trigonometry through the coordinate system

Think of it like this: instead of trying to prove the law using only geometric constructions (which is possible but more complex), we place the triangle in a coordinate system where we can use algebra and trigonometry together. It's like having both a map and GPS coordinates for the same location.

Step 1: Use the distance formula

Vertex C:

(0, 0)

Vertex A:

(b, 0)

Vertex B:

(a\cos C, a\sin C)

The distance between vertices A and B gives us the length of side c. The distance formula is: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ . Applying this to points A(b, 0) and B(a cos C, a sin C):

c^2 = (a\cos C - b)^2 + (a\sin C)^2

Step 2: Expand and simplify

We expand the squared terms and group like terms. Notice that we get terms involving $\cos^2 C$ and $\sin^2 C$ , which we can combine using the Pythagorean identity.

c^2 = a^2\cos^2 C - 2ab\cos C + b^2 + a^2\sin^2 C

c^2 = a^2(\cos^2 C + \sin^2 C) + b^2 - 2ab\cos C

Step 3: Apply the Pythagorean identity

The key insight: $\cos^2 C + \sin^2 C = 1$ by the Pythagorean identity. This is why the Law of Cosines is connected to the Pythagorean theorem—they both rely on this fundamental trigonometric relationship.

c^2 = a^2 + b^2 - 2ab\cos C

The result

This is a fundamental identity in geometry! The Law of Cosines generalizes the Pythagorean theorem to any triangle, connecting all three sides and one angle. It's essential for solving triangles, navigation, computer graphics, and many practical applications.

Why this proof method works

You might wonder: why not use pure geometry? The answer is that coordinate geometry gives us a systematic way to prove the law for all triangles, regardless of whether they're acute, right, or obtuse. Plus, it shows us the deep connection between geometry, algebra, and trigonometry—they're really different views of the same mathematical structure.

Other proof methods exist (using the Law of Sines, vector methods, or geometric constructions), but the coordinate geometry approach is particularly elegant because it shows us why the cosine term appears: it comes from the cross-term when we expand the distance formula.

Examples for beginners

Here are concrete examples showing how to use the Law of Cosines:

Example 1: Right triangle (Pythagorean theorem)

Given a right triangle with sides a = 3, b = 4, and angle C = 90°. Since $\cos(90°) = 0$ , the Law of Cosines gives: $c^2 = 3^2 + 4^2 - 2(3)(4)(0) = 9 + 16 = 25$ . Therefore, c = 5. This is the familiar 3-4-5 right triangle, and the Law of Cosines correctly reduces to the Pythagorean theorem.

Example 2: Acute triangle

Given a triangle with sides a = 5, b = 6, and angle C = 60°. Using the Law of Cosines: $c^2 = 5^2 + 6^2 - 2(5)(6)\cos(60°)$ . Since $\cos(60°) = 0.5$ , we get: $c^2 = 25 + 36 - 60(0.5) = 61 - 30 = 31$ . Therefore, c = √31 ≈ 5.57.

Example 3: Obtuse triangle

Given a triangle with sides a = 4, b = 5, and angle C = 120°. Since $\cos(120°) = -0.5$ , the Law of Cosines gives: $c^2 = 4^2 + 5^2 - 2(4)(5)(-0.5) = 16 + 25 + 20 = 61$ . Therefore, c = √61 ≈ 7.81. Notice how the negative cosine increases the value of c², reflecting that obtuse angles "stretch" the opposite side.

Example 4: Finding an angle

Given all three sides: a = 7, b = 8, c = 9. We can rearrange the Law of Cosines to find angle C: $\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{49 + 64 - 81}{2(7)(8)} = \frac{32}{112} = \frac{2}{7}$ . Therefore, C = arccos(2/7) ≈ 73.4°.

Common Mistakes

When working with Law of Cosines, students often encounter several common pitfalls. Understanding these mistakes can help you avoid them and apply the identity correctly.

Forgetting to check domain restrictions

Incorrect:

Applying Law of Cosines without verifying that all variables satisfy the required conditions (e.g., denominators not zero, square roots of non-negative numbers).

Correct approach:

Always check the domain of each variable before applying the identity. Verify that all conditions are met.

Why this matters: Many identities have implicit domain restrictions that must be satisfied for the identity to hold.

Order of operations errors

Incorrect:

Incorrectly applying operations when using Law of Cosines, especially with fractions or exponents.

Correct approach:

Follow the correct order of operations: parentheses, exponents, multiplication/division, addition/subtraction.

Why this matters: Order of operations is critical for correctly applying mathematical identities.

Quantum Implications

While the Law of Cosines is primarily geometric, it has important applications in physics, particularly in vector analysis and quantum mechanics:

In quantum mechanics, the Law of Cosines appears when analyzing the relationship between quantum states. The inner product of two quantum state vectors can be expressed using the Law of Cosines, relating the magnitude of their difference to their individual magnitudes and the angle between them in Hilbert space.

In vector mechanics, the Law of Cosines describes the relationship between the magnitudes of vectors and their dot product. If two vectors have magnitudes |a| and |b| and make angle θ, then the magnitude of their difference is: $|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta$ . This is exactly the Law of Cosines applied to vectors.

In quantum computing, when analyzing qubit states on the Bloch sphere, the Law of Cosines helps relate the angles between states and their overlap probabilities. The fidelity between quantum states can be expressed using cosine relationships derived from the Law of Cosines.

Quantum error correction codes use geometric relationships to protect quantum information. The Law of Cosines appears in the analysis of quantum error-correcting codes, where the geometric structure of the code space relates to the angles between error syndromes. This geometric perspective helps design more efficient error correction schemes that can protect quantum information from decoherence and noise.

In quantum state tomography, the process of reconstructing an unknown quantum state from measurements, the Law of Cosines helps relate the measurement outcomes to the geometric structure of the state space. The angles between measurement bases and the quantum state determine the probabilities of different measurement outcomes, and these relationships can be expressed using the Law of Cosines, enabling efficient reconstruction of quantum states from experimental data.

Philosophical Implications

The Law of Cosines raises questions about the relationship between mathematical geometry and physical space. Does the law describe physical reality, or is it a mathematical construct?

The mathematical realist view: The Law of Cosines describes a fundamental relationship that exists in physical space. Triangles in the real world (whether formed by light rays, forces, or physical objects) obey this law because space itself has this geometric structure.

The formalist view: The Law of Cosines is a consequence of our mathematical definitions and axioms. We define triangles, angles, and distances in a way that makes the law true. The elegance comes from our choice of mathematical framework.

The middle ground: While the geometric relationships may be discovered, the mathematical language (coordinates, trigonometry) is invented. The Law of Cosines exists in both realms—the underlying spatial truth is discovered, but its algebraic expression is crafted.

Patents and practical applications

Navigation

Calculating distances and bearings when two sides and the included angle are known. Used in GPS, maritime navigation, and aviation.

Surveying

Determining distances between points when direct measurement is impossible. Essential for mapping and construction.

Physics

Analyzing forces, vectors, and spatial relationships in mechanics. Used in statics, dynamics, and vector analysis.

Computer Graphics

Calculating angles and distances in 3D rendering, collision detection, and spatial queries. Fundamental to game engines and CAD software.

Engineering

Structural analysis and design of triangular frameworks. Used in truss design, bridge engineering, and architectural planning.

Robotics

Inverse kinematics and path planning require solving triangles using the Law of Cosines.

Astronomy

Calculating distances to stars using parallax measurements, where the Law of Cosines relates the observed angles and baseline distances.

Is it fundamental?

Is the Law of Cosines fundamental, or is it a natural generalization that reveals patterns in how we understand geometry? It generalizes the Pythagorean theorem to all triangles, unifying geometry and trigonometry, showing how angles and distances are connected. But is this connection fundamental to space itself, or fundamental to our description of spatial relationships?

The law appears in navigation (calculating distances and bearings), surveying (determining inaccessible distances), physics (vector analysis and mechanics), computer graphics (3D rendering and collision detection), and engineering (structural analysis). Yet we might ask: does this ubiquity suggest that spatial relationships are fundamental to reality, or does it reflect that the law provides a useful tool for our measurements and models?

The elegant reduction to the Pythagorean theorem when the angle is 90° reveals the law as a natural extension of geometric relationships. It shows that special cases (right triangles) are part of a more general pattern. But is this pattern fundamental to geometry, or a consequence of how we've chosen to measure and represent geometric relationships? The question of whether geometry is discovered or constructed remains debated.

Open questions and research frontiers

Non-Euclidean geometry

The Law of Cosines holds in Euclidean (flat) space. In curved spaces (spherical or hyperbolic geometry), there are analogous laws. Research explores how these generalizations relate to general relativity and the curvature of spacetime.

Higher dimensions

The law extends to higher-dimensional simplices (generalized triangles). Research in computational geometry explores these extensions for applications in data analysis and machine learning.

Quantum geometry

In quantum gravity theories, space itself may be discrete or quantized. How do geometric laws like the Law of Cosines manifest in such theories? This remains an active area of research.

Computational applications

Modern research explores efficient algorithms for triangle solving and applications in computer vision, robotics, and augmented reality, where the Law of Cosines is used for pose estimation and spatial reasoning.

People and milestones

Euclid (c. 300 BCE) provided the geometric foundation in his Elements. Islamic mathematicians like Al-Khwarizmi and al-Kāshī developed the trigonometric form during the Golden Age. Regiomontanus and Viète in Renaissance Europe further refined the law. Euler standardized modern trigonometric notation and identities.

Modern contributors include: Carl Friedrich Gauss, who used the law in surveying and geodesy; mathematicians developing non-Euclidean geometry, who found analogous laws in curved spaces; and computer scientists, who apply the law in computational geometry and graphics.

Related Identities

External References

Discovered Patterns

Research Notes

Loading notes...

Angle C (degrees): 60°

Side a: 3

Side b: 4

Side c = √(a² + b² - 2ab·cos(C)) = 3.61

Visualization Guide:

━ Blue triangle — The triangle outline with vertices A, B, and C. Side $a$ is opposite vertex A (connects vertices B and C), side $b$ is opposite vertex B (connects vertices A and C), and side $c$ is opposite vertex C (connects vertices A and B).

━ Green labels — Each side is labeled with its corresponding letter ( $a$ , $b$ , or $c$ ) at its midpoint. The labels show which side corresponds to which variable in the Law of Cosines formula $c^2 = a^2 + b^2 - 2ab\cos(C)$ .

━ Orange arc and label — The arc indicates angle C, which is the angle at vertex C (opposite side $c$ ). The orange label "C" marks this angle. This is the angle used in the formula $c^2 = a^2 + b^2 - 2ab\cos(C)$ .

━ White vertex labels — The vertices are labeled A, B, and C. Vertex C is where angle C is located, and side $c$ is opposite this vertex.

━ White axes — The coordinate system showing the triangle's position in 2D space. The triangle is centered at the origin for better visualization.

Controls: Adjust angle C and sides a and b using the sliders. Watch how side $c$ changes according to the Law of Cosines. When angle C is 90°, the triangle becomes a right triangle and the formula reduces to the Pythagorean theorem ( $c^2 = a^2 + b^2$ ).