Sine (sin) for Complex Numbers

Calculation of sin(z) - trigonometric function in the complex plane

Sine Calculator

Sine of Complex Numbers

The sine sin(z) of a complex number z = x + yi is calculated using real and hyperbolic functions. It is a periodic function with period 2π and can take arbitrarily large values (not bounded to [-1, 1]).

Angle z = x + yi (radians)
+
i
Calculation Result
sin(z) =
For purely real numbers: |sin(x)| ≤ 1, for complex numbers |sin(z)| can be > 1!

Sine - Properties

Formula for Complex Numbers
\[\sin(z) = \sin(x)\cosh(y) + i\cos(x)\sinh(y)\]

With z = x + yi

Euler's Formula
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]
Period 2π
Odd function sin(-z) = -sin(z)
Important Properties
  • Periodic with period 2π
  • Odd function: sin(-z) = -sin(z)
  • \(\sin^2(z) + \cos^2(z) = 1\) (Pythagorean identity)
  • Not bounded for complex z
Relations
  • \(\sin(z) = \cos(z - \pi/2)\)
  • \(\sin(2z) = 2\sin(z)\cos(z)\)
  • \(\sin(z \pm w) = \sin z \cos w \pm \cos z \sin w\)
  • \(\sinh(iz) = i\sin(z)\)


Formulas for Sine of Complex Numbers

The sine sin(z) of a complex number z = x + yi is calculated using a combination of trigonometric and hyperbolic functions.

Cartesian Form
\[\sin(x + yi) = \sin(x)\cosh(y) + i\cos(x)\sinh(y)\]

Real part: \(\sin(x)\cosh(y)\)
Imaginary part: \(\cos(x)\sinh(y)\)

Euler's Formula
\[\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]

Exponential representation

Step-by-Step Example

Calculation: sin(3 + 5i)
Step 1: Apply formula

z = 3 + 5i

x = 3 (real part)

y = 5 (imaginary part)

Step 2: Calculate real part

\(\text{Re} = \sin(3) \cdot \cosh(5)\)

\(= (0.14112) \cdot (74.20995)\)

\(\approx 10.473\)

Step 3: Calculate imaginary part

\(\text{Im} = \cos(3) \cdot \sinh(5)\)

\(= (-0.98999) \cdot (74.20321)\)

\(\approx -73.461\)

Step 4: Result

\(\sin(3 + 5i) = \text{Re} + i\text{Im}\)

\(\approx 10.473 - 73.461i\)

Observation

The magnitude \(|\sin(3 + 5i)| \approx 74.20\) is much greater than 1! This is typical for complex arguments with large imaginary parts, as cosh(y) and sinh(y) grow exponentially.

More Examples

Example 1: sin(0)

z = 0

\(\sin(0) = \sin(0)\cosh(0)\)

\(= 0 \cdot 1 = 0\)

Example 2: sin(π/2)

z = π/2 ≈ 1.5708

\(\sin(\pi/2) = \sin(\pi/2)\cosh(0)\)

\(= 1 \cdot 1 = 1\)

Example 3: sin(i)

z = i (purely imaginary)

\(\sin(i) = i\sinh(1)\)

\(\approx 1.175i\)

Example 4: sin(π)

z = π ≈ 3.1416

\(\sin(\pi) = \sin(\pi)\cosh(0)\)

\(\approx 0\)

Example 5: sin(1 + i)

z = 1 + i

\(\text{Re} = \sin(1)\cosh(1) \approx 1.298\)
\(\text{Im} = \cos(1)\sinh(1) \approx 0.635\)

\(\approx 1.298 + 0.635i\)

Example 6: sin(2i)

z = 2i (purely imaginary)

\(\sin(2i) = i\sinh(2)\)

\(\approx 3.627i\)

Sine - Detailed Description

Definition

The sine is one of the fundamental trigonometric functions.

For real numbers:
In a right triangle:
\[\sin(\alpha) = \frac{\text{Opposite}}{\text{Hypotenuse}}\]

Range: [-1, 1]
Period: 2π

For Complex Numbers

Calculation with z = x + yi:

\[\sin(z) = \sin(x)\cosh(y) + i\cos(x)\sinh(y)\]

• Real part: \(\sin(x)\cosh(y)\)
• Imaginary part: \(\cos(x)\sinh(y)\)
• Not bounded! Can become arbitrarily large

Important Properties

  • Periodicity: \(\sin(z + 2\pi) = \sin(z)\)
  • Odd function: \(\sin(-z) = -\sin(z)\)
  • Pythagorean identity: \(\sin^2(z) + \cos^2(z) = 1\)
  • Derivative: \(\frac{d}{dz}\sin(z) = \cos(z)\)

Addition Formulas

Sum formula:
\[\sin(z \pm w) = \sin z \cos w \pm \cos z \sin w\]
Double angle:
\[\sin(2z) = 2\sin(z)\cos(z)\]

Relations to Other Functions

• \(\sin(z) = \cos(z - \pi/2)\) (phase shift)
• \(\sinh(iz) = i\sin(z)\) (hyperbolic ↔ trigonometric)
• \(\sin(iz) = i\sinh(z)\) (inverse)
• \(e^{iz} = \cos(z) + i\sin(z)\) (Euler's formula)

Applications

Physics
  • Oscillations and waves
  • Alternating current
  • Harmonic oscillators
  • Acoustics
Geometry
  • Angle calculation
  • Circle functions
  • Projections
  • Rotations
Signal Processing
  • Fourier transformation
  • Frequency analysis
  • Waveforms
  • Filtering
Important Difference: Real vs. Complex
Real arguments (x ∈ ℝ):
  • Bounded: -1 ≤ sin(x) ≤ 1
  • Periodic with period 2π
  • Odd function
Complex arguments (z ∈ ℂ):
  • Not bounded! |sin(z)| can become arbitrarily large
  • Periodic with period 2π
  • Odd function
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