Law of Cosines Calculator: Free Online Triangle Solver

How to Use the Law of Cosines Calculator to Find Sides or Angles

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What is Law of Cosines Calculator?

A Law of Cosines Calculator helps you instantly solve for missing triangle sides or angles using the cosine rule. Provide any two sides and included angle, or all three sides, to compute unknown values with accuracy.

This tool works for any plane triangle, not just right triangles. Students, teachers, and professionals can quickly get solutions without tedious manual calculations, perfect for mobile and classroom use.

Formula Behind Law of Cosines Calculator

The law of cosines formula relates the sides and angles of a triangle: a² = b² + c² – 2bc·cosA. To find an angle, rearrange: cosA = (b² + c² – a²) / (2bc). By inputting known sides or angles, you can solve any plane triangle using these relationships.

Law of Cosines Example Table

Given	Find	Result
b = 7, c = 9, A = 50°	Side a	a ≈ 7.24
a = 7, b = 9, c = 12	Angles	A ≈ 34°, B ≈ 47°, C ≈ 99°
a = 5, b = 6, c = 7	Angles	A ≈ 44°, B ≈ 55°, C ≈ 81°

Steps to Use Law of Cosines Calculator

Choose whether you want to find a missing side or angle.
Enter known triangle values in the fields provided.
Click "Calculate" to get instant triangle solutions and steps.

Why Use Vedantu’s Law of Cosines Calculator?

This tool eliminates calculation mistakes and saves revision time. Get quick, stepwise explanations for every triangle scenario—anytime, anywhere.

With support for both SSS and SAS situations, Vedantu's calculator offers trusted accuracy for CBSE, ICSE, Olympiad, and competitive exam prep. Every solution matches curriculum standards and is reviewed by expert educators.

Applications of Law of Cosines Calculator

Solve real-world surveying, navigation, and engineering problems involving irregular triangles. The calculator makes these tasks efficient for students, teachers, and professionals.

It is invaluable for solving exam questions, classroom homework, and practical geometry, offering support beyond the basics. You’ll also find it helpful in advanced trigonometry and even certain physics problems. Check out topics like the Law of Sines or Pythagorean Theorem to expand your geometry knowledge, and see our guides on Area of Triangle or Trigonometry Concepts for deeper learning. For triangle basics, visit Geometry Basics on Vedantu.

FAQs on Law of Cosines Calculator: Free Online Triangle Solver

1. What is the Law of Cosines?

The Law of Cosines is a fundamental trigonometric formula that connects the lengths of the sides of a triangle to the cosine of one of its angles. It's incredibly useful for solving triangles where you know three sides (SSS) or two sides and the included angle (SAS). This law is a generalization of the Pythagorean theorem, extending its application beyond right-angled triangles.

2. What is the formula for the Law of Cosines?

The Law of Cosines can be expressed in three ways, depending on which angle you're focusing on. For a triangle with sides a, b, c and angles A, B, C opposite those sides respectively, the formulas are: a² = b² + c² - 2bc cosA; b² = a² + c² - 2ac cosB; c² = a² + b² - 2ab cosC. These formulas are crucial for calculating unknown side lengths or angles.

3. How do I use the Law of Cosines to find a missing side?

To find a missing side, you need to know the other two sides and the angle between them. Substitute the known values into the appropriate Law of Cosines formula (for example, if you want to find side 'a', use a² = b² + c² - 2bc cosA). Solve the equation for the unknown side length. Remember to consider the units of measurement (e.g., centimeters, meters).

4. How do I use the Law of Cosines to find a missing angle?

To find a missing angle, you need to know all three sides of the triangle. Rearrange the Law of Cosines formula to solve for the cosine of the angle. For example, to find angle A, use cosA = (b² + c² - a²) / (2bc). Then, find the inverse cosine (cos⁻¹) of the result to obtain the angle's measure in degrees or radians.

5. When should I use the Law of Cosines instead of the Law of Sines?

Use the Law of Cosines when you have the following information about a triangle: 1) Three sides (SSS) and you need to find the angles. 2) Two sides and the included angle (SAS) and you need to find the third side or the other angles. The Law of Sines is more suitable when you have different combinations of sides and angles (ASA, AAS, SSA).

6. What are some real-world applications of the Law of Cosines?

The Law of Cosines has many practical applications in various fields. It is used extensively in surveying to determine distances and angles, in navigation to calculate the distance between locations on Earth, in engineering for structural analysis and design, and in physics for various calculations involving vectors and triangles.

7. Can I use the Law of Cosines with obtuse triangles?

Yes, absolutely! The Law of Cosines works for all types of triangles—acute, right-angled, and obtuse. The formula remains the same regardless of the triangle's angles.

8. How accurate are the results from the Law of Cosines?

The accuracy of the results depends on the accuracy of the input values. Using precise measurements will lead to more accurate calculations. Rounding errors can occur during calculations, so it's best to use as many decimal places as possible during intermediate steps and round only at the final stage.

9. What if I get a negative value when calculating cosine?

If you obtain a negative value for cosine during your calculation, it simply indicates that the angle you are solving for is obtuse (greater than 90 degrees). Use your calculator's inverse cosine function (cos⁻¹) to find the angle's value. Make sure your calculator is set to the appropriate angle mode (degrees or radians).

10. Are there any limitations to using the Law of Cosines?

While the Law of Cosines is a powerful tool, it's crucial to have sufficient information to apply it correctly. You need at least three pieces of data (sides or angles) to solve for the unknowns. In the ambiguous case (SSA), you may need to use additional methods to determine the possible solutions.

11. How is the Law of Cosines related to the Pythagorean Theorem?

The Pythagorean Theorem is actually a special case of the Law of Cosines. When the angle A in a triangle is 90 degrees (a right-angled triangle), cosA becomes 0, and the Law of Cosines simplifies to a² = b² + c², which is the Pythagorean Theorem.