maths:circular

# a brief summary of circular functions and trigonometry

## 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 