Sin 2 Theta Calculator
Calculate sin(2θ) using the double angle formula and trigonometric identities
Calculate Sin(2θ)
Enter the angle θ to calculate sin(2θ)
Sin(2θ) Results
Formula Used
Method: Direct angle calculation
Formula: sin(2θ) = 2sin(θ)cos(θ)
Input: θ = 0°
Calculation: sin(2θ) = 2 × 0.000000 × 1.000000 = 0.000000
Step-by-Step Solution
- Step 1: Given angle θ = 0°
- Step 2: Convert to radians: θ = 0.000000 rad
- Step 3: Calculate sin(θ) = 0.000000
- Step 4: Calculate cos(θ) = 1.000000
- Step 5: Apply double angle formula: sin(2θ) = 2sin(θ)cos(θ)
- Step 6: sin(2θ) = 2 × 0.000000 × 1.000000 = 0.000000
Common sin(2θ) Values
| θ (degrees) | θ (radians) | sin(θ) | cos(θ) | sin(2θ) |
|---|---|---|---|---|
| 0° | 0 | 0.000 | 1.000 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/2 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3/2 |
| 90° | π/2 | 1.000 | 0.000 | 0 |
| 120° | 2π/3 | √3/2 | -1/2 | -√3/2 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/2 |
| 180° | π | 0.000 | -1.000 | 0 |
Double Angle Formulas
Sine
sin(2θ) = 2sin(θ)cos(θ)
Cosine
cos(2θ) = cos²(θ) - sin²(θ)
Tangent
tan(2θ) = 2tan(θ)/(1 - tan²(θ))
Alternative Forms
Using Sum and Difference
sin(2θ) = (sin(θ) + cos(θ))² - 1
Using Tangent
sin(2θ) = 2tan(θ)/(1 + tan²(θ))
Half-Angle Relation
sin(θ) = 2sin(θ/2)cos(θ/2)
Calculator Tips
Use angle input for direct calculation
Sin/Cos input validates Pythagorean identity
Tangent method handles all angle cases
Results include complementary functions
Understanding the Double Angle Formula: sin(2θ)
What is sin(2θ)?
The sine of a double angle, sin(2θ), is a fundamental trigonometric identity that expresses the sine of twice an angle in terms of the sine and cosine of the original angle. The most common form is sin(2θ) = 2sin(θ)cos(θ), which is derived from the compound angle formula.
Derivation
Starting with the compound angle formula:
sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
Setting α = β = θ:
sin(2θ) = 2sin(θ)cos(θ)
Alternative Forms
Standard Form
sin(2θ) = 2sin(θ)cos(θ)
Using Sum/Difference
sin(2θ) = (sin(θ) + cos(θ))² - 1
Using Tangent
sin(2θ) = 2tan(θ)/(1 + tan²(θ))
Applications
Wave Analysis
Double angle formulas are essential in analyzing wave patterns, frequency doubling, and harmonic motion
Engineering
Used in structural analysis, signal processing, and control systems for phase calculations
Calculus
Simplifies integration and differentiation of trigonometric functions
Key Properties
Domain and Range
- • Domain: All real numbers
- • Range: [-1, 1]
- • Period: π (half the period of sin(θ))
- • Amplitude: 1
Special Values
- • sin(2 × 0°) = sin(0°) = 0
- • sin(2 × 30°) = sin(60°) = √3/2
- • sin(2 × 45°) = sin(90°) = 1
- • sin(2 × 60°) = sin(120°) = √3/2