Cosine (cos) for Complex Numbers

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

Cosine Calculator

Cosine of Complex Numbers

The cosine cos(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)
+
Imaginary part (y)
i
Calculation Result
cos(z) =
For purely real numbers: |cos(x)| ≤ 1, for complex numbers |cos(z)| can be > 1!

Cosine - Properties

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

With z = x + yi

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


Formulas for Cosine of Complex Numbers

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

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

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

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

Exponential representation

Step-by-Step Example

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

z = 3 + 5i

x = 3 (real part)

y = 5 (imaginary part)

Step 2: Calculate real part

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

\(= (-0.98999) \cdot (74.20995)\)

\(\approx -73.47\)

Step 3: Calculate imaginary part

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

\(= -(0.14112) \cdot (74.20321)\)

\(\approx -10.47\)

Step 4: Result

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

\(\approx -73.47 - 10.47i\)

Observation

The magnitude \(|\cos(3 + 5i)| \approx 74.21\) 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: cos(0)

z = 0

\(\cos(0) = \cos(0)\cosh(0)\)

\(= 1 \cdot 1 = 1\)

Example 2: cos(π)

z = π ≈ 3.1416

\(\cos(\pi) = \cos(\pi)\cosh(0)\)

\(= -1 \cdot 1 = -1\)

Example 3: cos(i)

z = i (purely imaginary)

\(\cos(i) = \cos(0)\cosh(1)\)

\(= 1 \cdot \cosh(1) \approx 1.543\)

Example 4: cos(π/2)

z = π/2 ≈ 1.5708

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

\(\approx 0\)

Example 5: cos(1 + i)

z = 1 + i

\(\text{Re} = \cos(1)\cosh(1) \approx 0.833\)
\(\text{Im} = -\sin(1)\sinh(1) \approx -0.989\)

\(\approx 0.833 - 0.989i\)

Example 6: cos(2i)

z = 2i (purely imaginary)

\(\cos(2i) = \cosh(2)\)

\(\approx 3.762\)

Cosine - Detailed Description

Definition

The cosine is one of the fundamental trigonometric functions.

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

Range: [-1, 1]
Period: 2π

For Complex Numbers

Calculation with z = x + yi:

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

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

Important Properties

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

Addition Formulas

Sum formula:
\[\cos(z \pm w) = \cos z \cos w \mp \sin z \sin w\]
Double angle:
\[\cos(2z) = \cos^2(z) - \sin^2(z)\]
\[= 2\cos^2(z) - 1 = 1 - 2\sin^2(z)\]

Relations to Other Functions

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

Applications

Physics
  • Oscillations and waves
  • Alternating current
  • Fourier analysis
  • Quantum mechanics
Geometry
  • Angle calculation
  • Rotations
  • Polar coordinates
  • Vector projection
Signal Processing
  • Fourier transformation
  • Filter design
  • Modulation
  • Spectral analysis
Important Difference: Real vs. Complex
Real arguments (x ∈ ℝ):
  • Bounded: -1 ≤ cos(x) ≤ 1
  • Periodic with period 2π
  • Even function
Complex arguments (z ∈ ℂ):
  • Not bounded! |cos(z)| can become arbitrarily large
  • Periodic with period 2π
  • Even function
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