Law of Cosines Calculator That Solves for The Degrees
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Reviewed by Calculator Editorial Team
The Law of Cosines is a fundamental trigonometric identity that relates the lengths of the sides of a triangle to the cosine of one of its angles. This calculator solves for the angle in degrees when you know the lengths of all three sides of a triangle.
What is the Law of Cosines?
The Law of Cosines extends the Pythagorean theorem to any triangle, not just right-angled ones. It states that for any triangle with sides a, b, and c, and angle γ opposite side c, the following relationship holds:
Law of Cosines Formula
c² = a² + b² - 2ab cos(γ)
When solving for angle γ (in degrees):
γ = arccos[(a² + b² - c²) / (2ab)]
This formula allows you to find an angle when you know all three sides of a triangle. The Law of Cosines is particularly useful in navigation, surveying, engineering, and computer graphics.
How to Use This Calculator
- Enter the lengths of all three sides of your triangle (a, b, and c)
- Click the "Calculate" button
- View the calculated angle in degrees
- Review the step-by-step solution
- Use the visualization to understand the relationship between the sides and angle
Important Notes
- The sum of any two sides must be greater than the third side (triangle inequality theorem)
- All side lengths must be positive numbers
- The calculator automatically converts radians to degrees
The Formula
The Law of Cosines formula for solving angle γ is:
Angle Calculation Formula
γ = arccos[(a² + b² - c²) / (2ab)]
Where:
- γ = angle opposite side c (in degrees)
- a, b, c = lengths of the sides of the triangle
The arccos function returns the angle whose cosine is the specified value, and the result is automatically converted to degrees.
Worked Example
Let's solve for angle γ in a triangle with sides a=5, b=7, and c=6:
- Calculate the numerator: a² + b² - c² = 25 + 49 - 36 = 38
- Calculate the denominator: 2ab = 2 × 5 × 7 = 70
- Divide numerator by denominator: 38/70 = 0.5429
- Calculate arccos(0.5429) ≈ 57.1°
Therefore, angle γ is approximately 57.1 degrees.
Practical Applications
The Law of Cosines has numerous real-world applications including:
- Navigation and surveying
- Engineering and construction
- Computer graphics and animation
- Astronomy and celestial mechanics
- Robotics and motion planning
Understanding how to use the Law of Cosines calculator can help professionals in these fields make accurate measurements and calculations.
Frequently Asked Questions
What is the difference between the Law of Cosines and the Pythagorean theorem?
The Pythagorean theorem only applies to right-angled triangles, while the Law of Cosines works for any triangle. The Law of Cosines generalizes the Pythagorean theorem to all types of triangles.
Can I use this calculator for non-triangle shapes?
No, this calculator is specifically designed for triangular shapes. For other shapes, you would need a different type of calculator.
What if I enter side lengths that don't form a valid triangle?
The calculator will check if the side lengths satisfy the triangle inequality theorem (sum of any two sides must be greater than the third side). If not, it will display an error message.
How accurate are the calculations?
The calculations are performed using JavaScript's built-in Math functions, which provide accurate results to approximately 15 decimal places.