Cos 2θ Calculator
Calculate double angle trigonometric functions: cos(2θ), sin(2θ), and tan(2θ)
Calculate Double Angle Functions
Results
1.000000
cos(2θ) = 1.000000
1.000000
cos(2θ)
0.000000
sin(2θ)
0.000000
tan(2θ)
Original Angle (θ) Values
cos(θ): 1.000000
sin(θ): 0.000000
tan(θ): 0.000000
cos(2θ) Formula Verification
Formula 1: cos²(θ) - sin²(θ) = 1.000000
Formula 2: 2cos²(θ) - 1 = 1.000000
Formula 3: 1 - 2sin²(θ) = 1.000000
Step-by-Step Solution
1.Given: θ = 0°
2.Calculate: cos(2θ) = cos(2 × 0°) = cos(0°)
3.Method 1: cos(2θ) = cos²(θ) - sin²(θ)
4.cos(θ) = 1.000000, sin(θ) = 0.000000
5.cos(2θ) = (1.000000)² - (0.000000)² = 1.000000
6.Method 2: cos(2θ) = 2cos²(θ) - 1 = 2(1.000000)² - 1 = 1.000000
7.Method 3: cos(2θ) = 1 - 2sin²(θ) = 1 - 2(0.000000)² = 1.000000
8.Result: cos(2θ) = 1.000000
Double Angle Formulas
Cosine
cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = 2cos²(θ) - 1
cos(2θ) = 1 - 2sin²(θ)
Sine
sin(2θ) = 2sin(θ)cos(θ)
Tangent
tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Common Double Angles
cos(0°) = 1cos(0°) = 1
cos(60°) = 0.5cos(120°) = -0.5
cos(90°) = 0cos(180°) = -1
sin(30°) = 0.5sin(60°) = 0.866
sin(45°) = 0.707sin(90°) = 1
Understanding Double Angle Formulas
What are Double Angle Formulas?
Double angle formulas are trigonometric identities that express trigonometric functions of twice an angle (2θ) in terms of trigonometric functions of the original angle (θ). These formulas are essential tools in trigonometry and calculus.
cos(2θ) Formulas
The cosine double angle formula has three equivalent forms:
- •Form 1: cos(2θ) = cos²(θ) - sin²(θ)
- •Form 2: cos(2θ) = 2cos²(θ) - 1
- •Form 3: cos(2θ) = 1 - 2sin²(θ)
Derivation
The double angle formulas are derived from the angle addition formulas:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Setting A = B = θ:
cos(2θ) = cos(θ + θ)
cos(2θ) = cos(θ)cos(θ) - sin(θ)sin(θ)
cos(2θ) = cos²(θ) - sin²(θ)
Applications
- •Solving trigonometric equations
- •Integration and differentiation
- •Physics and engineering calculations
- •Signal processing and wave analysis
Example Applications
Physics Example
In wave interference, the amplitude of two waves with the same frequency can be calculated using double angle formulas when the phase difference is 2θ.
Engineering Example
In mechanical engineering, double angle formulas are used to analyze the motion of rotating machinery and calculate stress distributions.