107 LearnersLast updated on June 24th, 2025
Law Of Cosines Calculator
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.
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.
Mistake 1
Inputting incorrect side lengths or angles.
Ensure all input values are accurate. For example, measure sides and angles carefully to avoid errors in calculations.
Mistake 2
Using the wrong formula for the given situation.
Use the appropriate law of cosines formula based on what information you have and what you need to find.
Mistake 3
Forgetting to convert angle units.
Make sure to convert angles to the correct units (degrees or radians) that the calculator requires.
Mistake 4
Misinterpreting the results.
Understand what the calculator output represents, whether it's a side length or an angle, and ensure it fits the context of your problem.
Mistake 5
Confusing the triangle's sides and angles.
Ensure that each side and angle corresponds correctly to its opposite in the triangle, as mismatches can lead to incorrect results.
Law Of Cosines Calculator Examples
Problem 1
Find the length of side c in a triangle with sides a = 5, b = 7, and angle C = 60 degrees.
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.
Problem 2
Determine the angle C in a triangle with sides a = 8, b = 10, and c = 12.
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.
Problem 3
Calculate side a in a triangle with sides b = 9, c = 11, and angle A = 45 degrees.
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.
Problem 4
Find the angle B in a triangle with sides a = 6, b = 8, and c = 10.
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.
Problem 5
Determine side b in a triangle with sides a = 7, c = 5, and angle B = 30 degrees.
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.
FAQs on Using the Law Of Cosines Calculator
1.How do you use the law of cosines to find a side?
2.Can the law of cosines determine angles?
3.Why is the law of cosines useful?
4.What units should angles be in for calculations?
5.Is the law of cosines calculator accurate?
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.
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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
