Tangent (tan) for Complex Numbers

Calculation of tan(z) - ratio of sine to cosine

Tangent Calculator

Tangent of Complex Numbers

The tangent tan(z) is the ratio of sine to cosine: \(\tan(z) = \frac{\sin(z)}{\cos(z)}\). For complex numbers it is a periodic function with period π and has poles at (2k+1)π/2.

Angle z = x + yi (radians)
+
i
Calculation Result
tan(z) =
Caution: tan(z) has poles at z = (2k+1)π/2, k ∈ ℤ!

Tangent - Properties

Formula for Complex Numbers
\[\tan(z) = \frac{\sin(2x)}{\cos(2x)+\cosh(2y)} + i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}\]

With z = x + yi

Quotient Representation
\[\tan(z) = \frac{\sin(z)}{\cos(z)}\]
Period π
Odd function tan(-z) = -tan(z)
Important Properties
  • Periodic with period π
  • Odd function: tan(-z) = -tan(z)
  • Poles at z = (2k+1)π/2
  • \(1 + \tan^2(z) = \frac{1}{\cos^2(z)}\)
Relations
  • \(\tan(z) = \frac{1}{\cot(z)}\)
  • \(\tan(2z) = \frac{2\tan(z)}{1-\tan^2(z)}\)
  • \(\tan(z \pm w) = \frac{\tan z \pm \tan w}{1 \mp \tan z \tan w}\)
  • \(\tanh(iz) = i\tan(z)\)

Formulas for Tangent of Complex Numbers

The tangent tan(z) of a complex number z = x + yi is the ratio of sine to cosine and combines trigonometric with hyperbolic functions.

Cartesian Form
\[\tan(z) = \frac{\sin(2x)}{\cos(2x)+\cosh(2y)} + i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}\]

Real part: \(\frac{\sin(2x)}{\cos(2x)+\cosh(2y)}\)
Imaginary part: \(\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}\)

Quotient Representation
\[\tan(z) = \frac{\sin(z)}{\cos(z)}\]

Ratio of sine to cosine

Step-by-Step Example

Calculation: tan(0.3 + 0.5i)
Step 1: Apply formula

z = 0.3 + 0.5i

x = 0.3, y = 0.5

2x = 0.6, 2y = 1.0

Step 2: Calculate denominator

\(\cos(0.6) + \cosh(1.0)\)

\(= 0.825 + 1.543\)

\(\approx 2.368\)

Step 3: Calculate real part

\(\text{Re} = \frac{\sin(0.6)}{2.368}\)

\(= \frac{0.565}{2.368}\)

\(\approx 0.238\)

Step 4: Calculate imaginary part

\(\text{Im} = \frac{\sinh(1.0)}{2.368}\)

\(= \frac{1.175}{2.368}\)

\(\approx 0.496\)

Step 5: Result

\(\tan(0.3 + 0.5i) = \text{Re} + i\text{Im}\)

\(\approx 0.238 + 0.496i\)

More Examples

Example 1: tan(0)

z = 0

\(\tan(0) = \frac{\sin(0)}{\cos(0)}\)

\(= \frac{0}{1} = 0\)

Example 2: tan(π/4)

z = π/4 ≈ 0.7854

\(\tan(\pi/4) = \frac{\sin(\pi/4)}{\cos(\pi/4)}\)

\(= \frac{\sqrt{2}/2}{\sqrt{2}/2} = 1\)

Example 3: tan(i)

z = i (purely imaginary)

\(\tan(i) = i\tanh(1)\)

\(\approx 0.762i\)

Example 4: tan(π/6)

z = π/6 ≈ 0.5236

\(\tan(\pi/6) = \frac{1}{\sqrt{3}}\)

\(\approx 0.577\)

Example 5: tan(π/2) - Pole!

z = π/2 ≈ 1.5708

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

Undefined (pole)!

Example 6: tan(1 + i)

z = 1 + i

\(\approx 0.272 + 1.084i\)

Tangent - Detailed Description

Definition

The tangent is the ratio of sine to cosine.

Quotient representation:
\[\tan(z) = \frac{\sin(z)}{\cos(z)}\]

For real numbers:
In a right triangle:
\[\tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}}\]
Range: (-∞, ∞)
Period: π

For Complex Numbers

Calculation with z = x + yi:

\[\tan(z) = \frac{\sin(2x)}{\cos(2x)+\cosh(2y)} + i\frac{\sinh(2y)}{\cos(2x)+\cosh(2y)}\]

• Not bounded
• Poles at (2k+1)π/2

Important Properties

  • Periodicity: \(\tan(z + \pi) = \tan(z)\)
  • Odd function: \(\tan(-z) = -\tan(z)\)
  • Poles: at \(z = \frac{(2k+1)\pi}{2}\)
  • Derivative: \(\frac{d}{dz}\tan(z) = \frac{1}{\cos^2(z)}\)

Addition Formulas

Sum formula:
\[\tan(z \pm w) = \frac{\tan z \pm \tan w}{1 \mp \tan z \tan w}\]
Double angle:
\[\tan(2z) = \frac{2\tan(z)}{1-\tan^2(z)}\]

Relations to Other Functions

• \(\tan(z) = \frac{1}{\cot(z)}\) (reciprocal of cotangent)
• \(\tanh(iz) = i\tan(z)\) (hyperbolic ↔ trigonometric)
• \(\tan(iz) = i\tanh(z)\) (inverse)
• \(1 + \tan^2(z) = \frac{1}{\cos^2(z)} = \sec^2(z)\)

Poles

Caution: Singularities!

The tangent has poles (singularities) where the denominator becomes zero:

\[\tan(z) \text{ undefined for } z = \frac{(2k+1)\pi}{2}, \quad k \in \mathbb{Z}\]

Examples:
• tan(π/2) → ∞ (undefined)
• tan(3π/2) → ∞ (undefined)
• tan(-π/2) → -∞ (undefined)

Applications

Geometry
  • Slope calculation
  • Angle determination
  • Inclination angle
  • Triangle calculations
Physics
  • Projectile trajectory
  • Optics (refraction)
  • Oscillations
  • Phase analysis
Engineering
  • Road construction (gradients)
  • Navigation
  • Geodesy
  • Structural engineering
Period: π instead of 2π

Unlike sine and cosine, the tangent has a period of π (not 2π):

\[\tan(z + \pi) = \tan(z)\]

Reason: \(\tan(z + \pi) = \frac{\sin(z+\pi)}{\cos(z+\pi)} = \frac{-\sin(z)}{-\cos(z)} = \frac{\sin(z)}{\cos(z)} = \tan(z)\)
Both numerator and denominator change sign!


More complex functions

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Airy function • Derivative Airy function •
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