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maths:circular

a brief summary of circular functions and trigonometry

Introduction

Circular functions:

  • NB. π radians = 180 degrees.
  • unit circle defined by x2+y2 = 1 and each point of the circle represented by (cost,sint) where t = angle in radians polar coordinates which is also twice the area of the sector subtended by that angle
    • ie. cos(t)2+sin(t)2 = 1
  • 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;

Hyperbolic functions

  • a point (x,y) on the hyperbolic curve x2-y2 = 1 can be given by (cosh(t),sinh(t)) where
    • t = twice the area subtended by the angle from the origin, the x axis, and the curve (analogous to circular functions)
    • cosh(t)2-sinh(t)2 = 1
    • cosh(t) = (et+e-t)/2
    • sinh(t) = (et-e-t)/2
    • a catenary curve as on suspension bridges: y := coshx
maths/circular.txt · Last modified: 2021/09/25 15:58 by gary1