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.
Arcsin - Properties
Formula
With complex logarithm and square root
Definition
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)
- \(\arcsin(\sin(z)) = z + 2\pi k\)
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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
With \(\ln\) = complex logarithm
Alternative Form
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.
\[\sin(\arcsin(z)) = z\]
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(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:
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)
In a right triangle:
\[\sin(\alpha) = \frac{\text{Opposite}}{\text{Hypotenuse}}\]
The arc sine calculates the angle α from this ratio:
\[\alpha = \arcsin\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)\]
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
Example: 0.6435 rad ≈ 36.87°
Degrees → Radians
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)
= 0
= π/4 = 45°
= π/3 = 60°
= 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
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