Law of Cosines Calculator
Use the Law of Cosines to find unknown sides or angles in a triangle. Supports SAS (Side-Angle-Side) and SSS (Side-Side-Side) cases. Get step-by-step solutions with detailed explanations and interactive triangle visualization!
About Law of Cosines Calculator
Welcome to our Law of Cosines Calculator, a comprehensive online tool for solving triangles using the Law of Cosines. Whether you're dealing with SAS (Side-Angle-Side) or SSS (Side-Side-Side) cases, our calculator provides accurate solutions with detailed step-by-step explanations and interactive visualizations.
What is the Law of Cosines?
The Law of Cosines (also known as the Cosine Rule) is a fundamental theorem in trigonometry that generalizes the Pythagorean theorem. It relates the sides of a triangle to the cosine of one of its angles. The law states:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$
where $a$, $b$, and $c$ are the lengths of the sides of a triangle, and $C$ is the angle opposite side $c$. Similarly, we can write:
$$a^2 = b^2 + c^2 - 2bc\cos(A)$$
$$b^2 = a^2 + c^2 - 2ac\cos(B)$$
When rearranged to solve for angles, the Law of Cosines becomes:
$$\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$$
Key Features of Our Law of Cosines Calculator
- Multiple Case Support: Solve SAS and SSS triangle cases.
- Dual Angle Units: Works with both degrees and radians for maximum flexibility.
- Step-by-Step Solutions: Detailed explanations of each calculation step.
- Interactive Visualization: Graphical representation of the solved triangle.
- Triangle Inequality Check: Validates SSS inputs to ensure a valid triangle exists.
- Comprehensive Validation: Ensures all inputs are valid with meaningful error messages.
- Educational Tool: Perfect for students learning trigonometry and triangle geometry.
Understanding Triangle Cases
SAS (Side-Angle-Side)
When you know two sides and the angle between them, you can use the Law of Cosines to find the third side, then calculate the remaining angles.
- Given: Sides $a$ and $b$, and the angle $C$ between them
- Find: Side $c$, angles $A$ and $B$
- Solution: Unique (one solution)
- Formula: $c^2 = a^2 + b^2 - 2ab\cos(C)$
SSS (Side-Side-Side)
When you know all three sides of a triangle, you can use the Law of Cosines to find all three angles.
- Given: Sides $a$, $b$, and $c$
- Find: Angles $A$, $B$, and $C$
- Solution: Unique (one solution, if triangle exists)
- Important: The triangle inequality must be satisfied: the sum of any two sides must be greater than the third side
How to Use the Law of Cosines Calculator
- Select the case type: Choose SAS or SSS from the dropdown menu.
- Choose angle unit: Select degrees or radians.
- Enter your values:
- SAS: Input 1 = Side a, Input 2 = Side b, Input 3 = Angle C (the angle between sides a and b)
- SSS: Input 1 = Side a, Input 2 = Side b, Input 3 = Side c
- Click "Calculate" to get your solution.
- Review the step-by-step solution and triangle visualization.
Applications of the Law of Cosines
The Law of Cosines is widely used in various fields:
- Navigation: Determining distances and bearings in triangular paths.
- Astronomy: Calculating distances between celestial objects.
- Surveying: Land measurement and mapping applications.
- Engineering: Structural analysis, force calculations, and design.
- Physics: Vector analysis and resultant force calculations.
- Architecture: Roof design, structural support calculations.
- Computer Graphics: 3D modeling, collision detection, and rendering.
- Robotics: Path planning and inverse kinematics.
Mathematical Background
The Law of Cosines is a generalization of the Pythagorean theorem. When the angle $C = 90°$ (a right angle), $\cos(C) = 0$, and the formula reduces to:
$$c^2 = a^2 + b^2$$
This is exactly the Pythagorean theorem! The Law of Cosines works for all types of triangles: acute, obtuse, and right triangles.
Derivation
The Law of Cosines can be derived using coordinate geometry or by applying the Pythagorean theorem to a triangle that has been divided by an altitude.
Important Properties
- The sum of angles in any triangle is always $180°$ (or $\pi$ radians)
- In any triangle, the largest side is opposite the largest angle
- For the SSS case, the triangle inequality must hold: $a + b > c$, $a + c > b$, and $b + c > a$
- The Law of Cosines can be used to determine the type of triangle:
- If $c^2 < a^2 + b^2$, the triangle is acute
- If $c^2 = a^2 + b^2$, the triangle is right-angled
- If $c^2 > a^2 + b^2$, the triangle is obtuse
Why Use Our Law of Cosines Calculator?
- Accuracy: Precise calculations with proper rounding and error handling.
- Educational: Learn through detailed step-by-step solutions.
- Time-Saving: Get instant results instead of manual calculations.
- Comprehensive: Handles both SAS and SSS cases with complete validation.
- Visual Learning: See the triangle you're solving with our visualization feature.
- Free and Accessible: No registration required, use anytime, anywhere.
- Mobile Friendly: Works on all devices - desktop, tablet, and mobile.
Law of Cosines vs. Law of Sines
While both laws are used to solve triangles, they have different applications:
- Law of Cosines: Best for SAS and SSS cases
- Law of Sines: Best for ASA, AAS, and SSA (ambiguous) cases
- The Law of Cosines is more computationally stable for small angles
- Together, these laws can solve any triangle given sufficient information
Additional Resources
For more information on the Law of Cosines and triangle geometry, check out these resources:
Reference this content, page, or tool as:
"Law of Cosines Calculator" at https://MiniWebtool.com// from MiniWebtool, https://MiniWebtool.com/
by miniwebtool team. Updated: Nov 18, 2025
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