Circular Functions

• circular functions
• inverse circular functions
• see also:
  • Algebra
  • Mensuration
  • Calculus
  • Coordinate geometry, parabolae, etc
  • Finance equations
  • Logarithmic, exponential and hyperbolic functions
  • Probability theory

Circular functions:

• NB. Pi radians = 180 degrees.
• tanx=sinx/cosx, cosx not= 0;
• cotx=cosx/sinx;
• secx=1/cosx;
• cosecx=1/sinx;
• 1+(tan²)x = (sec²)x, cosx not= 0;
• 1+(cot²)x = (cosec²)x, sinx not= 0;
• (cos²)x + (sin²)x = 1;
• Where x is in degrees:
  • sin(90-x) = cosx; cos(90-x) = sinx;
  • tan(90-x) = cotx; cot(90-x) = tanx;
  • sin(180-x)= sinx; cos(180-x)=-cosx;
  • tan(180-x)=-tanx; cot(180-x)=-cotx;
  • sin(180+x)=-sinx; cos(180+x)=-cosx;
  • tan(180+x)= tanx; cot(180+x)= cotx;
  • sin(360-x)=-sinx; cos(360-x)= cosx;
  • tan(360-x)=-tanx; cot(360-x)=-cotx;
  • sin(-x) =-sinx; cos(-x) = cosx;
  • tan(-x) =-tanx; cot(-x) =-cotx;
  • sin(x+y) = sinxcosy + cosxsiny;
  • sin(x-y) = sinxcosy - cosxsiny;
  • cos(x+y) = cosxcosy - sinxsiny;
  • cos(x-y) = cosxcosy + sinxsiny;
  • tan(x+y) = (tanx + tany)/(1 - tanxtany);
  • tan(x-y) = (tanx - tany)/(1 + tanxtany);
  • sin2x = 2sinxcosx = 2tanx/(1 + (tan^2)x);
  • cos2x = (cos²)x - (sin²)x = 2(cos²)x - 1;
    • = 1-2(sin²)x = (1-(tan²)x)/(1+(tan²)x);
  • tan2x = 2tanx/(1 - (tan²)x);
  • 2sinxcosx = sin(x+y) + sin(x-y);
  • 2cosxcosy = cos(x+y) + cos(x-y);
  • 2sinxsiny = cos(x-y) - cos(x+y);
  • sinx + siny = 2sin((x+y)/2)cos((x-y)/2);
  • sinx - siny = 2cos((x+y)/2)sin((x-y)/2);
  • cosx + cosy = 2cos((x+y)/2)cos((x-y)/2);
  • cosx - cosy =-2sin((x+y)/2)sin((x-y)/2);

Inverse Circular Functions:

• NB. Must restrict circular function to make 1-to-1:
  • ie. sinx, xE[-90,90]; cosx, xE[0,180]; tanx, -90<x<90;
• General Solutions:
  • sinx = a, => x = 180n + ((-1)ⁿ)Arsina;
  • cosx = a, => x = 360n +/- Arcosa;
  • tanx = a, => x = 180n + Artana;