Bounds

 |sin(x)| <= 1,|cos(x)| <= 1,|sec(x)| >= 1,|csc(x)| >= 1.

Identities

 sec(x) = 1/cos(x),csc(x) = 1/sin(x),cot(x) = 1/tan(x),tan(x) = sin(x)/cos(x),cot(x) = cos(x)/sin(x).sin(-x) = -sin(x),cos(-x) = cos(x),tan(-x) = -tan(x),cot(-x) = -cot(x),sec(-x) = sec(x),csc(-x) = -csc(x).sin(/2-x) = cos(x),cos(/2-x) = sin(x),tan(2-x) = cot(x),cot(/2-x) = tan(x),sec(/2-x) = csc(x),csc(/2-x) = sec(x).sin(/2+x) = cos(x),cos(/2+x) = -sin(x),tan(/2+x) = -cot(x),cot(/2+x) = -tan(x),sec(/2+x) = -csc(x),csc(/2+x) = sec(x).sin(-x) = sin(x),cos(-x) = -cos(x),tan(-x) = -tan(x),cot(-x) = -cot(x),sec(-x) = -sec(x),csc(-x) = csc(x).sin(+x) = -sin(x),cos(+x) = -cos(x),tan(+x) = tan(x),cot(+x) = cot(x),sec(+x) = -sec(x),csc(+x) = -csc(x).sin(2+x) = sin(x),cos(2+x) = cos(x),tan(2+x) = tan(x),cot(2+x) = cot(x),sec(2+x) = sec(x),csc(2+x) = csc(x).sin2(x) + cos2(x) = 1,tan2(x) + 1 = sec2(x),1 + cot2(x) = csc2(x).sin(x+y) = sin(x)cos(y) + cos(x)sin(y),cos(x+y) = cos(x)cos(y) - sin(x)sin(y),tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)],cot(x+y) = [cot(x)cot(y)-1]/[cot(x)+cot(y)].sin(x-y) = sin(x)cos(y) - cos(x)sin(y),cos(x-y) = cos(x)cos(y) + sin(x)sin(y),tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)],cot(x-y) = [cot(x)cot(y)+1]/[cot(y)-cot(x)].sin(2x) = 2 sin(x)cos(x),cos(2x) = cos2(x) - sin2(x),= 2 cos2(x) - 1,= 1 - 2 sin2(x),tan(2x) = [2 tan(x)]/[1-tan2(x)],cot(2x) = [cot2(x)-1]/[2 cot(x)].|sin(x/2)| = sqrt([1-cos(x)]/2),|cos(x/2)| = sqrt([1+cos(x)]/2),|tan(x/2)| = sqrt([1-cos(x)]/[1+cos(x)]),tan(x/2) = [1-cos(x)]/sin(x),= sin(x)/[1+cos(x)].sin(3x) = 3 sin(x) - 4 sin3(x),cos(3x) = 4 cos3(x) - 3 cos(x),tan(3x) = [3 tan(x)-tan3(x)]/[1-3 tan2(x)].sin(4x) = 4 sin(x)cos(x)[2 cos2(x)-1],cos(4x) = 8 cos4(x) - 8 cos2(x) + 1.sin(5x) = 5 sin(x) - 20 sin3(x) + 16 sin5(x),cos(5x) = 16 cos5(x) - 20 cos3(x) + 5 cos(x).sin(6x) = 2 sin(x)cos(x)[16 cos4(x) - 16 cos2(x) + 3],cos(6x) = 32 cos6(x) - 48 cos4(x) + 18 cos2(x) - 1.sin(nx) = 2 sin([n-1]x)cos(x) - sin([n-2]x),cos(nx) = 2 cos([n-1]x)cos(x) - cos([n-2]x),tan(nx) = (tan[(n-1)x]+tan[x])/(1-tan[(n-1)x]tan[x]).sin(x)cos(y) = [sin(x+y) + sin(x-y)]/2,cos(x)sin(y) = [sin(x+y) - sin(x-y)]/2,cos(x)cos(y) = [cos(x-y) + cos(x+y)]/2,sin(x)sin(y) = [cos(x-y) - cos(x+y)]/2.sin(x) + sin(y) = 2 sin[(x+y)/2]cos[(x-y)/2],sin(x) - sin(y) = 2 cos[(x+y)/2]sin[(x-y)/2],cos(x) + cos(y) = 2 cos[(x+y)/2]cos[(x-y)/2],cos(x) - cos(y) = -2 sin[(x+y)/2]sin[(x-y)/2],tan(x) + tan(y) = sin(x+y)/[cos(x)cos(y)],tan(x) - tan(y) = sin(x-y)/[cos(x)cos(y)],cot(x) + cot(y) = sin(x+y)/[sin(x)sin(y)],cot(x) - cot(y) = -sin(x-y)/[sin(x)sin(y)].[sin(x)+sin(y)]/[cos(x)+cos(y)] = tan[(x+y)/2],[sin(x)-sin(y)]/[cos(x)+cos(y)] = tan[(x-y)/2],[sin(x)+sin(y)]/[cos(x)-cos(y)] = -cot[(x-y)/2],[sin(x)-sin(y)]/[cos(x)-cos(y)] = -cot[(x+y)/2],[sin(x)+sin(y)]/[sin(x)-sin(y)] = tan[(x+y)/2]/tan[(x-y)/2].sin2(x) - sin2(y) = sin(x+y)sin(x-y),cos2(x) - cos2(y) = -sin(x+y)sin(x-y),cos2(x) - sin2(y) = cos(x+y)cos(x-y).sin2(x) = (1 - cos[2x])/2,cos2(x) = (1 + cos[2x])/2,tan2(x) = (1 - cos[2x])/(1 + cos[2x]),sin3(x) = (3 sin[x] - sin[3x])/4,cos3(x) = (3 cos[x] + cos[3x])/4,sin4(x) = (3 - 4 cos[2x] + cos[4x])/8,cos4(x) = (3 + 4 cos[2x] + cos[4x])/8,sin5(x) = (10 sin[x] - 5 sin[3x] + sin[5x])/16,cos5(x) = (10 cos[x] + 5 cos[3x] + cos[5x])/16,sin6(x) = (10 - 15 cos[2x] + 6 cos[4x] - cos[6x])/32,cos6(x) = (10 + 15 cos[2x] + 6 cos[4x] + cos[6x])/32,

Relations in Right Triangles

 /2

 A + B = /2,c2 = a2 + b2,sin(A) = cos(B) = a/c,cos(A) = sin(B) = b/c,tan(A) = cot(B) = a/b,cot(A) = tan(B) = b/a,sec(A) = csc(B) = c/b,csc(A) = sec(B) = c/a,ha = b,hb = a,hc = ab/c.

Case I: You are given a and A.

B =

/2 - A, c = a csc(A), b = a cot(A). 

Case II: You are given a and B.

A =

/2 - B, c = a sec(B), b = a tan(B). 

Case III: You are given c and A.

B =

/2 - A, a = c sin(A), b = c cos(A). 

Case IV: You are given a and b.

tan(A) = a/b, B =

/2 - A, c = a csc(A). 

Case V: You are given a and c.

sin(A) = a/c, B =

/2 - A, b = a cot(A). 

Relations in Oblique Triangles

 A + B + C = ,s = (a+b+c)/2, half the perimeter,r = radius of inscribed circle,R = radius of circumscribed circle,K = area.

 a/sin(A) = b/sin(B) = c/sin(C) = 2R.

 a2 = b2 + c2 - 2bc cos(A),b2 = c2 + a2 - 2ca cos(B),c2 = a2 + b2 - 2ab cos(C).

 (a+b)/(a-b) = tan[(A+B)/2]/tan[(A-B)/2],(b+c)/(b-c) = tan[(B+C)/2]/tan[(B-C)/2],(c+a)/(c-a) = tan[(C+A)/2]/tan[(C-A)/2].

 (a+b)/c = cos[(A-B)/2]/sin(C/2),(b+c)/a = cos[(B-C)/2]/sin(A/2),(c+a)/b = cos[(C-A)/2]/sin(B/2).

 (a-b)/c = sin[(A-B)/2]/cos(C/2),(b-c)/a = sin[(B-C)/2]/cos(A/2),(c-a)/b = sin[(C-A)/2]/cos(B/2).

 a = b cos(C) + c cos(B), b = c cos(A) + a cos(C), c = a cos(B) + b cos(A). tan[(A-B)/2] = [(a-b)/(a+b)]cot(C/2), tan[(B-C)/2] = [(b-c)/(b+c)]cot(A/2), tan[(C-A)/2] = [(c-a)/(c+a)]cot(B/2). sin(A) = 2K/(bc), sin(B) = 2K/(ca), sin(C) = 2K/(ab). K = sr = sqrt[s(s-a)(s-b)(s-c)], K = aha/2 = bhb/2 = chc/2, K = ab sin(C)/2 = bc sin(A)/2 = ca sin(B)/2, K = a2 sin(B)sin(C)/[2 sin(A)], = b2 sin(C)sin(A)/[2 sin(B)], = c2 sin(A)sin(B)/[2 sin(C)]. R = abc/(4K) = a/[2 sin(A)] = b/[2 sin(B)] = c/[2 sin(C)], r = K/s, = sqrt[(s-a)(s-b)(s-c)/s], = c sin(A/2)sin(B/2)/cos(C/2), = ab sin(C)/(2s), = (s-c)tan(C/2). sin(A/2) = sqrt[(s-b)(s-c)/(bc)], sin(B/2) = sqrt[(s-c)(s-a)/(ca)], sin(C/2) = sqrt[(s-a)(s-b)/(ab)]. cos(A/2) = sqrt[s(s-a)/(bc)], cos(B/2) = sqrt[s(s-b)/(ca)], cos(C/2) = sqrt[s(s-c)/(ab)]. tan(A/2) = sqrt[(s-b)(s-c)/{s(s-a)}] = r/(s-a), tan(B/2) = sqrt[(s-c)(s-a)/{s(s-b)}] = r/(s-b), tan(C/2) = sqrt[(s-a)(s-b)/{s(s-c)}] = r/(s-c). (a+b)/(a-b) = [sin(A)+sin(B)]/[sin(A)-sin(B)] = cot(C/2)/tan[(A-B)/2], (b+c)/(b-c) = [sin(B)+sin(C)]/[sin(B)-sin(C)] = cot(A/2)/tan[(B-C)/2], (c+a)/(c-a) = [sin(C)+sin(A)]/[sin(C)-sin(A)] = cot(B/2)/tan[(C-A)/2]. ha = a sin(B)sin(C)/sin(B+C) = a/[cot(B)+cot(C)] = b sin(C) = c sin(B), hb = b sin(C)sin(A)/sin(C+A) = b/[cot(C)+cot(A)] = c sin(A) = a sin(C), hc = c sin(A)sin(B)/sin(A+B) = c/[cot(A)+cot(B)] = a sin(B) = b sin(A).  cos(A) + cos(B) + cos(C) = 1 + r/R.