Cosine Law Calculator

The cosine law, also known as the law of cosines, is a fundamental trigonometric principle used to solve triangles. Whether you’re a student tackling geometry homework or an engineer working on complex designs, our Cosine Law Calculator simplifies the process of finding unknown sides and angles in triangles.

What is the Cosine Law?

The cosine law relates the lengths of the sides of a triangle to the cosine of one of its angles. It’s particularly useful when dealing with non-right triangles where the Pythagorean theorem doesn’t apply.

The formula for the cosine law is:

c² = a² + b² - 2ab cos(C)

Where:

a, b, and c are the lengths of the triangle’s sides
C is the angle opposite the side c

How to Use Our Cosine Law Calculator

Choose the type of triangle problem you’re solving:
- SSS (Side-Side-Side): When you know all three side lengths
- SAS (Side-Angle-Side): When you know two sides and the included angle
Enter the known values:
- For SSS: Input the lengths of all three sides
- For SAS: Input two side lengths and the angle between them
Click “Calculate” to get your results instantly
The calculator will display the unknown angles or side length, depending on your input

Examples of Cosine Law Calculations

Example 1: SSS Triangle

Given:

Side a = 5 cm
Side b = 7 cm
Side c = 8 cm

To find angle C:

cos(C) = (a² + b² - c²) / (2ab) cos(C) = (5² + 7² - 8²) / (2 _ 5 _ 7) C ≈ 82.82°

Example 2: SAS Triangle

Given:

Side a = 10 m
Side b = 12 m
Angle C = 60°

To find side c:

c² = a² + b² - 2ab cos(C) c² = 10² + 12² - 2 _ 10 _ 12 * cos(60°) c ≈ 10.39 m

Applications of the Cosine Law

The cosine law has numerous real-world applications:

Surveying and mapping
Navigation and GPS systems
Architecture and construction
Physics and engineering
Astronomy and space exploration

Tips for Using the Cosine Law Calculator

Always double-check your input values for accuracy
Remember that angles are typically in degrees for most calculators
Use the calculator to verify your manual calculations in homework or projects
Explore different triangle scenarios to deepen your understanding of trigonometry

Ready to solve your triangle problems? Try our Cosine Law Calculator now and experience the ease of trigonometry calculations at your fingertips!

Frequently Asked Questions

When should I use the cosine law instead of the sine law?

Use the cosine law when you have either all three sides (SSS) or two sides and the included angle (SAS) of a triangle. The sine law is better for AAS (Angle-Angle-Side) or ASA (Angle-Side-Angle) triangles.

Can the cosine law be used for right triangles?

Yes, the cosine law works for all triangles, including right triangles. However, for right triangles, the Pythagorean theorem is often simpler to use.

How accurate is the Cosine Law Calculator?

Our calculator uses precise mathematical algorithms and provides results accurate to several decimal places, suitable for most practical applications.

Can I use this calculator for 3D problems?

The cosine law is primarily for 2D triangles. For 3D problems, you might need to break down the geometry into 2D components or use more advanced tools.

Is there a way to solve triangles without a calculator?

While it's possible to solve triangles manually using logarithm tables or approximation methods, a calculator greatly simplifies the process and improves accuracy.

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Cosine Law Calculator