Law Of Cosines Calculator
2026-02-28 13:59 Diff
https://brightchamps.com/en-us/math/calculators/law-of-cosines-calculator

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Last updated on August 5, 2025

Calculators are reliable tools for solving simple mathematical problems and advanced calculations like trigonometry. Whether you’re cooking, tracking BMI, or planning a construction project, calculators will make your life easy. In this topic, we are going to talk about Law of Cosines calculators.

What is a Law Of Cosines Calculator?

A Law of Cosines calculator is a tool to determine the unknown side or angle of a triangle using the law of cosines. This law relates the lengths of the sides of a triangle to the cosine of one of its angles, making it a vital tool for solving triangles when you know two sides and the included angle or all three sides. The calculator simplifies these trigonometric calculations, making them quicker and more convenient.

How to Use the Law Of Cosines Calculator?

Given below is a step-by-step process on how to use the calculator:

Step 1: Enter the known values: Input the known sides and/or angles into the given fields.

Step 2: Click on calculate: Click on the calculate button to solve for the unknown side or angle.

Step 3: View the result: The calculator will display the result instantly.

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How to Apply the Law of Cosines?

The law of cosines states that for any triangle with sides a, b, and c, and angle C opposite side c: c² = a² + b² - 2ab * cos(C)

This formula is used to solve for side c if you know sides a, b, and angle C.

Similarly, if you know all sides, you can solve for an angle using the rearranged formula: cos(C) = (a² + b² - c²) / (2ab) The calculator applies these formulas to find the unknown side or angle as needed.

Tips and Tricks for Using the Law of Cosines Calculator

When using a Law of Cosines calculator, there are a few tips and tricks that can help you make the most of it:

Ensure your angle measurements are in the correct unit, either degrees or radians, as required by the calculator.

Double-check that you have the correct measurements for sides a, b, and c. Remember that the law of cosines is especially useful for non-right triangles.

Use the calculator to verify your manual calculations. Keep in mind that the law of cosines can also be used in reverse to find angles when all sides are known.

Common Mistakes and How to Avoid Them When Using the Law Of Cosines Calculator

We may think that when using a calculator, mistakes will not happen. But it is possible for errors to occur even when using a calculator.

Problem 1

Find the length of side c in a triangle with sides a = 5, b = 7, and angle C = 60 degrees.

Okay, lets begin

Use the formula: c² = a² + b² - 2ab * cos(C)

c² = 5² + 7² - 2 * 5 * 7 * cos(60)

c² = 25 + 49 - 70 * 0.5

c² = 74 - 35

c² = 39

c = √39 ≈ 6.24

Therefore, side c is approximately 6.24 units.

Explanation

By applying the law of cosines formula to the given values, we solved for the unknown side c, using the cosine of 60 degrees which equals 0.5.

Well explained 👍

Problem 2

Determine the angle C in a triangle with sides a = 8, b = 10, and c = 12.

Okay, lets begin

Use the formula: cos(C) = (a² + b² - c²) / (2ab)

cos(C) = (8² + 10² - 12²) / (2 * 8 * 10)

cos(C) = (64 + 100 - 144) / 160

cos(C) = 20 / 160

cos(C) = 0.125

C = cos⁻¹(0.125) ≈ 82.82 degrees

Therefore, angle C is approximately 82.82 degrees.

Explanation

Using the rearranged law of cosines formula, we calculated angle C by finding the inverse cosine of 0.125.

Well explained 👍

Problem 3

Calculate side a in a triangle with sides b = 9, c = 11, and angle A = 45 degrees.

Okay, lets begin

Use the formula: a² = b² + c² - 2bc * cos(A)

a² = 9² + 11² - 2 * 9 * 11 * cos(45)

a² = 81 + 121 - 198 * 0.7071

a² = 202 - 139.8078

a² = 62.1922

a = √62.1922 ≈ 7.88

Therefore, side a is approximately 7.88 units.

Explanation

By applying the law of cosines formula with the given angle A and sides b and c, we solved for side a.

Well explained 👍

Problem 4

Find the angle B in a triangle with sides a = 6, b = 8, and c = 10.

Okay, lets begin

Use the formula: cos(B) = (a² + c² - b²) / (2ac)

cos(B) = (6² + 10² - 8²) / (2 * 6 * 10)

cos(B) = (36 + 100 - 64) / 120

cos(B) = 72 / 120

cos(B) = 0.6 B = cos⁻¹(0.6) ≈ 53.13 degrees

Therefore, angle B is approximately 53.13 degrees.

Explanation

Using the law of cosines, we calculated angle B by finding the inverse cosine of 0.6.

Well explained 👍

Problem 5

Determine side b in a triangle with sides a = 7, c = 5, and angle B = 30 degrees.

Okay, lets begin

Use the formula: b² = a² + c² - 2ac * cos(B)

b² = 7² + 5² - 2 * 7 * 5 * cos(30)

b² = 49 + 25 - 70 * 0.866

b² = 74 - 60.62

b² = 13.38 b = √13.38 ≈ 3.66

Therefore, side b is approximately 3.66 units.

Explanation

Applying the law of cosines with the given angle B and sides a and c, we solved for side b.

Well explained 👍

FAQs on Using the Law Of Cosines Calculator

1.How do you use the law of cosines to find a side?

Input the known sides and angle into the formula c² = a² + b² - 2ab * cos(C), then solve for the unknown side.

2.Can the law of cosines determine angles?

Yes, by rearranging the formula to cos(C) = (a² + b² - c²) / (2ab), you can solve for angles if all sides are known.

3.Why is the law of cosines useful?

The law of cosines is useful for solving triangles that are not right triangles, especially when two sides and the included angle or all three sides are known.

4.What units should angles be in for calculations?

Make sure to use the unit (degrees or radians) that your calculator is set to handle, to avoid errors in calculations.

5.Is the law of cosines calculator accurate?

The calculator provides accurate results based on the input values and the law of cosines formula, so ensure your inputs are correct.

Glossary of Terms for the Law Of Cosines Calculator

  • Law of Cosines: A formula that relates the lengths of the sides of a triangle to the cosine of one of its angles.
  • Cosine: A trigonometric function that represents the adjacent side over the hypotenuse in a right-angled triangle.
  • Inverse Cosine: A function used to determine the angle whose cosine value is known.
  • Radian: A unit of angle measurement where the angle is defined by the arc length on a unit circle.
  • Non-right Triangle: A triangle that does not have a 90-degree angle, often solved using the law of cosines.

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables