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.
Arccos - Properties
Formula
With complex logarithm and square root
Definition
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)
- \(\arccos(\cos(z)) = z + 2\pi k\)
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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
With \(\ln\) = complex logarithm
Alternative Form
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.
\[\cos(\arccos(z)) = z\]
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(1) = 0
• arccos(0) = π/2 ≈ 1.5708
• arccos(-1) = π ≈ 3.1416
For Complex Numbers
For complex numbers, arccos is multivalued:
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)
In a right triangle:
\[\cos(\alpha) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\]
The arc cosine calculates the angle α from this ratio:
\[\alpha = \arccos\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right)\]
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
Example: 1.266 rad ≈ 72.54°
Degrees → Radians
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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