Law of Cosines
Find the third side of any triangle
Interactive Sandbox
Adjust values in real-time and observe mathematical cause-and-effect
Bounded Oscillatory Function: Sine and cosine oscillate smoothly between -1 and +1 with period 360° (2π rad).
Variable Mechanics
Roles, constraints & impacts
Side a
Formula Parameter
Side b
Formula Parameter
Angle C
Formula Parameter
At a Glance
Gotchas, units & real-world use
Formula Nature & Mechanics
Bounded Oscillatory Function: Sine and cosine oscillate smoothly between -1 and +1 with period 360° (2π rad).
Common Mistakes & Traps
- Using Degree mode when Radians are expected (or vice versa).
- Tangent asymptotes at 90° and 270° (division by cos θ = 0 causes undefined infinity).
- Forgetting inverse trigonometric domain limits (e.g., arcsin input must be between -1 and 1).
Dimensional Analysis & Units
Ratio of side lengths → Dimensionless Scalar [-1, +1]
Real-World & Industry Application
Audio sound wave synthesis, AC alternating current electrical waveforms, robotic arm joint kinematics, and astronomical navigation.
Worked Example: Law of Cosines
Given:
- a = 5
- b = 7
- C = 60
- c² = 5² + 7² - 2(5)(7)cos(60°)
- c² = 25 + 49 - 70(0.5)
- c² = 74 - 35
- c² = 39
- c = √39 = 6.24
Answer: 6.24
Frequently Asked Questions
What is the Law of Cosines formula?
c² = a² + b² - 2ab cos(C). Find the third side of any triangle.
How do I calculate Law of Cosines?
Enter Side a (a), Side b (b), Angle C (C) into the calculator. It applies c² = a² + b² - 2ab cos(C) and shows every step of the working.
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