Jump to content

Trigonometric constants expressed in real radicals

From Wikipedia, the free encyclopedia
This is an old revision of this page, as edited by Vespristiano (talk | contribs) at 06:22, 21 May 2004 (Boldened the subject). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radial forms which allow further simplification.

All values of sin/cos/tan of angles with 3 degree increments are derivable using identity expressions and values for 30 and 45 degrees.

Uses for constants

For example, the volume of a dodecahedron is:

  • V = 5*e3*cos(36)/tan2(36)
  • Using:
    • cos(36) = ($5+1)/4
    • tan(36) = $(5-2*$5)
  • It can be be simplified to:
    • V = e3*(15+7*sqrt(5))/4

Expressions not unique

Simplifying recursive radial expressions is nontrivial. The expressions here may look different from other variations.

Example:

  • sin(18) = [sqrt(3-sqrt(5)/2)]/2 = (sqrt(5)-1)/4
    • It's not obvious visually that this simplification is equivalent.

Table of constants

Shortcut: $(x) = square_root(x) = sqrt(x)

  • sin(0) = 0
  • cos(0) = 1
  • tan(0) = 0
  • sin(3) = [2*$(5+$5)*(1-$3)+$2*($5-1)*($3+1)]/16
  • cos(3) = [2*$(5+$5)*(1+$3)+$2*($5-1)*($3-1)]/16
  • tan(3) = [(2-$3)*(3+$5)-2]*(2-$(2*(5-$5)))/4
  • sin(6) = [$(6*(5-$5))- ($5+1)]/8
  • cos(6) = [$(2*(5-$5))+$3*($5-1)]/8
  • tan(6) = [$(5-2*$5)*($5+1)+$3*(1-$5)]/2
  • sin(9) = [-2*$(5-$5)+$2*($5+1)]/8
  • cos(9) = [ 2*$(5-$5)+$2*($5+1)]/8
  • tan(9) = -$(5-2*$5)*(2+$5)+($5+1)
  • sin(12) = [$(2*(5+$5))-$3*($5-1)]/8
  • cos(12) = [$(6*(5+$5))+ ($5-1)]/8
  • tan(12) = [$(5-2*$5)*(2+$5)+($5+1)]/2
  • sin(15) = $2*($3-1)/4
  • cos(15) = $2*($3+1)/4
  • tan(15) = 2-$3
  • cotan(15) = 2+$3
  • sin(18) = ($5-1)/4
  • cos(18) = $(2*(5+$5))/4
  • tan(18) = $(5*(5-2*$5))
  • cotan(18) = $(5+2*$5)
  • sin(21) = [2*$(5-$5)*($3+1)-$2*($3-1)*(1+$5)]/16
  • cos(21) = [2*$(5-$5)*($3-1)+$2*($3+1)*(1+$5)]/16
  • tan(21) = [$(5-2*$5)*(1+2*$3-$5)+(2+$3)*($5-3)+2]/2
  • sin(22.5) = $(2-$2)/2
  • cos(22.5) = $(2+$2)/2
  • tan(22.5) = $2-1
  • sin(24) = $(2*(5+$5))*(1-$5)+2*$3*(1+$5))/16
  • cos(24) = $(6*(5+$5))*($5-1)+2*(1+$5))/16
  • tan(24) = ?
  • cotan(24) = (($(2*5+$5)+2*$3)*($5-1))/4
  • sin(27) = ((2*$(5+$5)+$2(1-$5))/8
  • cos(27) = ((2*$(5+$5)+$2($5-1))/8
  • tan(27) = -$(5-2($5))+($5-1)
  • sin(30) = 1/2
  • cos(30) = $3/2
  • tan(30) = $3/3
  • sin(33) = (2*$(5+$5)*(-1+$3)+$2*($5-1)*(1+$3))/16
  • cos(33) = (2*$(5+$5)*(+1+$3)+$2*($5-1)*(1-$3))/16
  • tan(33) = ($(5(5-2*$5))*(-15+10*$3-7*$5+4*$15)+5*((-2+$3)*(3+$5)+2))/10
  • sin(36) = $(2*(5-$5))/4
  • cos(36) = ($5+1)/4
  • tan(36) = $(5-2*$5)
  • sin(39) = (2*$(5-$5)*(1-$3)+$2*(+1+$3)*(1+$5))/16
  • cos(39) = (2*$(5-$5)*(1+$3)+$2*(-1+$3)*(1+$5))/16
  • tan(39) = ($(2*(5+$5))-2)*((2-$3)*(-3+$5)+2)/4
  • sin(42) = ($(6*(5-$5))*(1+$5)+2*(1-$5))/16
  • cos(42) = ($(2*(5-$5))*(1+$5)+2*$3*(-1+$5))/16
  • tan(42) = (-$(5-2*$5)*(3+$5)+$3*(1+$5))/2
  • sin(45) = $2/2
  • cos(45) = $2/2
  • tan(45) = 1

See also

Retrieved from "https://en.wikipedia.org/w/index.php?title=Trigonometric_constants_expressed_in_real_radicals&oldid=3669643"