Arc Sine (arcsin) for Complex Numbers

Calculation of arcsin(z) - the inverse function of sine

Arcsin Calculator

Arc Sine (arcsin)

The arc sine arcsin(z) is the inverse function of sine: If \(\sin(w) = z\), then \(w = \arcsin(z)\). For complex numbers, the function is multivalued and has infinitely many values.

Sine value z = a + bi
+
Imaginary part (b)
i
Calculation Result
arcsin(z) (principal value) =
The function is multivalued: All values are \(w + 2\pi k\) with \(k \in \mathbb{Z}\)

Arcsin - Properties

Formula
\[\arcsin(z) = -i\ln\left(iz + \sqrt{1-z^2}\right)\]

With complex logarithm and square root

Definition
\[\sin(\arcsin(z)) = z\]
Real numbers [-1, 1] → [-π/2, π/2]
Complex Multivalued
Important Properties
  • Inverse function of sin(z)
  • Multivalued: \(w + 2\pi k, k \in \mathbb{Z}\)
  • Principal value: \(\text{Re}(w) \in [-\pi/2, \pi/2]\)
  • \(\arcsin(-z) = -\arcsin(z)\) (odd function)
Relations
  • \(\arcsin(z) + \arccos(z) = \frac{\pi}{2}\)
  • \(\arcsin(z) = \frac{\pi}{2} - \arccos(z)\)
  • \(\sin(\arcsin(z)) = z\) (definition)
  • \(\arcsin(\sin(z)) = z + 2\pi k\)

Formulas for Arc Sine of Complex Numbers

The arc sine arcsin(z) is the inverse function of sine and is defined by the complex logarithm.

Main Formula
\[\arcsin(z) = -i\ln\left(iz + \sqrt{1-z^2}\right)\]

With \(\ln\) = complex logarithm

Alternative Form
\[\arcsin(z) = \frac{\pi}{2} - \arccos(z)\]

Relation to arccos

Arc Sine - Detailed Description

Definition and Meaning

The arc sine (also arcsin or asin) is the inverse function of the sine function.

Definition:
\[\sin(\arcsin(z)) = z\]
The arc sine returns the angle (in radians)
whose sine has the value z.

Notation:
arcsin(z), asin(z), or \(\sin^{-1}(z)\)

For Real Numbers

For real numbers \(x \in [-1, 1]\):

Range:

\[\arcsin(x) \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\]

• arcsin(1) = π/2 ≈ 1.5708 rad = 90°
• arcsin(0) = 0
• arcsin(-1) = -π/2 ≈ -1.5708 rad = -90°

For Complex Numbers

For complex numbers, arcsin is multivalued:

Multivaluedness:
If \(w = \arcsin(z)\), then
\[w + 2\pi k \quad (k \in \mathbb{Z})\]
and
\[\pi - w + 2\pi k \quad (k \in \mathbb{Z})\]
are also valid solutions.

Principal value:
The principal value has \(\text{Re}(w) \in [-\pi/2, \pi/2]\)

Important Relations

  • \(\arcsin(z) + \arccos(z) = \frac{\pi}{2}\)
  • \(\arcsin(-z) = -\arcsin(z)\) (odd function)
  • \(\arcsin(\overline{z}) = \overline{\arcsin(z)}\)
  • \(\arcsin(z) = -i\ln(iz + \sqrt{1-z^2})\)

Caution

For complex z, \(|\arcsin(z)|\) can become arbitrarily large!
The function is only real for \(|z| \leq 1\).
For \(|z| > 1\), arcsin(z) is complex.

Geometric Meaning (Real Numbers)

Right Triangle:
In a right triangle:
\[\sin(\alpha) = \frac{\text{Opposite}}{\text{Hypotenuse}}\]
The arc sine calculates the angle α from this ratio:
\[\alpha = \arcsin\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)\]
Example:
Opposite: a = 3
Hypotenuse: c = 5
\[\sin(\alpha) = \frac{3}{5} = 0.6\]
\[\alpha = \arcsin(0.6) \approx 0.6435 \text{ rad}\]
\[\alpha \approx 36.87°\]

Conversion Radians ↔ Degrees

Radians → Degrees
\[\text{Degrees} = \frac{\text{Radians} \cdot 180°}{\pi}\]

Example: 0.6435 rad ≈ 36.87°

Degrees → Radians
\[\text{Radians} = \frac{\text{Degrees} \cdot \pi}{180°}\]

Example: 30° = π/6 ≈ 0.5236 rad

Calculation Examples

Example 1: arcsin(0.5)

Real number: z = 0.5

\(\arcsin(0.5) = \frac{\pi}{6}\)

≈ 0.5236 rad = 30°

Example 2: arcsin(1)

Maximum: z = 1

\(\arcsin(1) = \frac{\pi}{2}\)

≈ 1.5708 rad = 90°

Example 3: arcsin(-1)

Minimum: z = -1

\(\arcsin(-1) = -\frac{\pi}{2}\)

≈ -1.5708 rad = -90°

Example 4: arcsin(0.4 + 0.3i)

Complex number: z = 0.4 + 0.3i

Use formula:

\(\arcsin(z) = -i\ln(iz + \sqrt{1-z^2})\)

Result: see calculator above

Example 5: arcsin(2)

Outside [-1,1]: z = 2

\(\arcsin(2) = -i\ln(2i + \sqrt{-3})\)

≈ 1.571 - 1.317i (complex!)

Example 6: arcsin(i)

Imaginary unit: z = i

\(\arcsin(i) = -i\ln(i^2 + \sqrt{1-i^2})\)

≈ 0 + 0.881i

Special Values (real)
arcsin(0)
= 0
arcsin(√2/2)
= π/4 = 45°
arcsin(√3/2)
= π/3 = 60°
arcsin(1)
= π/2 = 90°

Applications

Geometry
  • Angle calculation in triangles
  • Projection on unit circle
  • Trajectories and paths
  • Coordinate transformations
Physics
  • Oscillations and waves
  • Pendulum motion
  • Projectile motion (launch angle)
  • Optics (Snell's law)
Mathematics
  • Complex analysis
  • Integral calculus
  • Differential equations
  • Fourier series

More complex functions

Absolute value (abs) • Angle • Conjugate • Division • Exponent • Logarithm to base 10 • Multiplication • Natural logarithm • Polarform • Power • Root • Reciprocal • Square root •
Cosh • Sinh • Tanh •
Acos • Asin • Atan • Cos • Sin • Tan •
Airy function • Derivative Airy function •
Bessel-I • Bessel-Ie • Bessel-J • Bessel-Je • Bessel-K • Bessel-Ke • Bessel-Y • Bessel-Ye •