Shunyaya Symbolic Mathematics — Inverse trigonometric functions (2.33)

Abstract
We lift arcsin, arccos, and arctan to symbolic numerals (m, a) as magnitude maps with alignment carry on their principal domains. Order claims follow classical monotonicity on those domains. Under collapse phi(m,a) = m, all formulas reduce to the classical inverse trigonometric functions.

Definitions (principal branches, alignment carry)

For a symbolic input x = (m, a):

arcsin*(m, a) = ( arcsin(m) , a )     with  m in [-1, 1], range in [-pi/2, +pi/2]
arccos*(m, a) = ( arccos(m) , a )     with  m in [-1, 1], range in [ 0,     pi ]
arctan*(m, a) = ( arctan(m) , a )     with  m in (-inf, +inf), range in (-pi/2, +pi/2)
  • Alignment is carried unchanged because these are unary magnitude maps on monotone domains.
  • Principal values are used throughout; any nonprincipal branches must be declared explicitly in the manifest.

Domains, ranges, and monotonicity

arcsin

  • Domain: [-1, 1]
  • Range: [-pi/2, +pi/2]
  • Monotonicity: strictly increasing on [-1, 1]
  • Order rule: on [-1,1], magnitude order is preserved under arcsin*.

arccos

  • Domain: [-1, 1]
  • Range: [0, pi]
  • Monotonicity: strictly decreasing on [-1, 1]
  • Order rule: on [-1,1], magnitude order is reversed under arccos*.

arctan

  • Domain: (-inf, +inf)
  • Range: (-pi/2, +pi/2)
  • Monotonicity: strictly increasing on R
  • Order rule: on R, magnitude order is preserved under arctan*.

Endpoints and continuity.

  • arcsin(±1) = ±pi/2 (continuous on [-1,1]).
  • arccos(±1) = {0, pi} (continuous on [-1,1]).
  • arctan(m) is continuous on R and asymptotes at ±pi/2 as m -> ±inf.
    Alignment carry leaves a bounded; no extra guard is needed for a.

Collapse, ordering, and ties

Collapse check.

phi( arcsin*(m, a) ) = arcsin(m)
phi( arccos*(m, a) ) = arccos(m)
phi( arctan*(m, a) ) = arctan(m)

Order rules (magnitude channel).

  • If f is increasing on its domain D: (m1,a1) <=_m (m2,a2) ⇒ f*(m1,a1) <=_m f*(m2,a2)
  • If f is decreasing on D (arccos), the inequality reverses.

Ties.
If magnitudes map to the same value (e.g., equal inputs), break ties by your declared symbolic preorder (e.g., via S_beta) rather than by ad hoc alignment rules.

Composition notes (principal branches)

  • sin*( arcsin*(m,a) ) = ( m , a ) for m in [-1,1].
  • cos*( arccos*(m,a) ) = ( m , a ) for m in [-1,1].
  • tan*( arctan*(m,a) ) = ( m , a ) for all real m.

Reverse compositions require domain/range restrictions: e.g.,

  • arcsin*( sin*(m,a) ) = ( arcsin( sin(m) ) , a ) equals (m,a) only if m in [-pi/2, +pi/2].
  • arccos*( cos*(m,a) ) equals (m,a) only if m in [0, pi].
  • arctan*( tan*(m,a) ) equals (m,a) only if m in (-pi/2, +pi/2).

Declare any broader periodic lifting policy in the manifest if needed.

Guards and error policies (manifest)

  • Out-of-domain input for arcsin*/arccos* (i.e., |m| > 1):
    • domain_guard = error (default; leave undefined), or
    • domain_guard = clamp (evaluate at m_clamped = sign(m) * min(|m|,1)), or
    • domain_guard = nan (return sentinel).
      State the policy explicitly on the page.
  • Numerical stability.
    Near |m| ≈ 1, use high-accuracy routines or arctan2-based formulas where appropriate. Alignment needs no special treatment (carried unchanged).

Worked examples

A) arcsin with preserved order.
x1 = (0.2, +0.6), x2 = (0.6, -0.3)

arcsin*(x1) = ( arcsin(0.2), +0.6 ) ≈ ( 0.2014 , +0.6 )
arcsin*(x2) = ( arcsin(0.6), -0.3 ) ≈ ( 0.6435 , -0.3 )

Since 0.2 < 0.6, magnitudes preserve order under arcsin.

B) arccos with reversed order.
y1 = ( -0.8 , +0.2 ), y2 = ( -0.4 , +0.9 )

arccos*(y1) ≈ ( 2.4981 , +0.2 )
arccos*(y2) ≈ ( 1.9823 , +0.9 )

Because arccos is decreasing, the larger input magnitude (-0.4 > -0.8) yields a smaller output magnitude.

C) arctan on R.
z1 = ( -2.0 , -0.7 ), z2 = ( 3.0 , +0.1 )

arctan*(z1) ≈ ( -1.1071 , -0.7 )
arctan*(z2) ≈ (  1.2490 , +0.1 )

As |m| grows, outputs approach ±pi/2 in magnitude; alignment is carried.

Implementation notes (manifest)

  • Function & domain: record branch (principal), domain intervals, and domain_guard.
  • Alignment policy: unary inverse trig lifts carry a; any composite products/ratios around them use M2.
  • Precision: for inputs |m| close to 1, prefer library implementations with correct rounding; document precision targets.
  • Zero policy: if results are exactly zero in magnitude, canonicalize to (0, +1).

Takeaway

Inverse trig functions lift cleanly by alignment carry on their principal domains. Monotonicity gives order-preservation for arcsin* and arctan*, and order-reversal for arccos*. Clear domain guards and branch declarations keep the lift collapse-safe; under collapse, everything is exactly classical.

Navigation
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.