Arc Cosine (arccos) for Complex Numbers

Calculation of arccos(z) - the inverse function of cosine

Arccos Calculator

Arc Cosine (arccos)

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

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

Arccos - Properties

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

With complex logarithm and square root

Definition
\[\cos(\arccos(z)) = z\]
Real numbers [-1, 1] → [0, π]
Complex Multivalued
Important Properties
  • Inverse function of cos(z)
  • Multivalued: \(w + 2\pi k, k \in \mathbb{Z}\)
  • Principal value: \(\text{Re}(w) \in [0, \pi]\)
  • \(\arccos(-z) = \pi - \arccos(z)\)
Relations
  • \(\arccos(z) + \arcsin(z) = \frac{\pi}{2}\)
  • \(\arccos(z) = \frac{\pi}{2} - \arcsin(z)\)
  • \(\cos(\arccos(z)) = z\) (definition)
  • \(\arccos(\cos(z)) = z + 2\pi k\)

Formulas for Arc Cosine of Complex Numbers

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

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

With \(\ln\) = complex logarithm

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

Relation to arcsin

Arc Cosine - Detailed Description

Definition and Meaning

The arc cosine (also arccos or acos) is the inverse function of the cosine function.

Definition:
\[\cos(\arccos(z)) = z\]
The arc cosine returns the angle (in radians)
whose cosine has the value z.

Notation:
arccos(z), acos(z), or \(\cos^{-1}(z)\)

For Real Numbers

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

Range:

\[\arccos(x) \in [0, \pi]\]

• arccos(1) = 0
• arccos(0) = π/2 ≈ 1.5708
• arccos(-1) = π ≈ 3.1416

For Complex Numbers

For complex numbers, arccos is multivalued:

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

Principal value:
The principal value has \(\text{Re}(w) \in [0, \pi]\)

Important Relations

  • \(\arccos(z) + \arcsin(z) = \frac{\pi}{2}\)
  • \(\arccos(z) = \frac{\pi}{2} - \arcsin(z)\)
  • \(\cos(\arccos(z)) = z\) (definition)
  • \(\arccos(\cos(z)) = z + 2\pi k\)

Caution

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

Geometric Meaning (Real Numbers)

Right Triangle:
In a right triangle:
\[\cos(\alpha) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]
The arc cosine calculates the angle α from this ratio:
\[\alpha = \arccos\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)\]
Example:
Adjacent: b = 6
Hypotenuse: c = 20
\[\cos(\alpha) = \frac{6}{20} = 0.3\]
\[\alpha = \arccos(0.3) \approx 1.266 \text{ rad}\]
\[\alpha \approx 72.54°\]

Conversion Radians ↔ Degrees

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

Example: 1.266 rad ≈ 72.54°

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

Example: 90° = π/2 ≈ 1.5708 rad

Calculation Examples

Example 1: arccos(0.5)

Real number: z = 0.5

\(\arccos(0.5) = \frac{\pi}{3}\)

≈ 1.047 rad = 60°

Example 2: arccos(1)

Maximum: z = 1

\(\arccos(1) = 0\)

= 0 rad = 0°

Example 3: arccos(-1)

Minimum: z = -1

\(\arccos(-1) = \pi\)

≈ 3.142 rad = 180°

Example 4: arccos(0.4 + 0.3i)

Complex number: z = 0.4 + 0.3i

Use formula:

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

Result: see calculator above

Example 5: arccos(2)

Outside [-1,1]: z = 2

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

≈ 0 - 1.317i (complex!)

Example 6: arccos(i)

Imaginary unit: z = i

\(\arccos(i) = \frac{\pi}{2} - i\ln(1+\sqrt{2})\)

≈ 1.571 - 0.881i

Applications

Geometry
  • Angle calculation in triangles
  • Determining vector angles
  • Dot product applications
  • 3D geometry
Physics
  • Wave mechanics
  • Oscillations
  • Electrical engineering (impedance)
  • Optics (refraction angle)
Mathematics
  • Complex analysis
  • Integral calculus
  • Differential equations
  • Fourier transformation

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Airy function • Derivative Airy function •
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