Page 42 - Standard Handbook Of Petroleum & Natural Gas Engineering

Trigonometry 3 1

1 + tan2x = sec2x
1 + cotzx = CSCZX
sin(-x) = -sin x, cos(-x) = cos x, tan(-x) = -tan x
Two Angles
sin(x + y) = sin x cos y + cos x sin y
sin(x - y) = sin x cos y - cos x sin y
cos(x + y) = cos x cos y - sin x sin y
cos(x - y) = cos x cos y + sin x sin y
tan(x + y) = (tan x + tan y)/(l - tan x tan y)
tan(x - y) = (tan x - tan y)/(l + tan x tan y)
cot(x + y) = (cot x cot y - l)/(cot y + cot x)
cot(x - y) = (cot x cot y + l)/(cot y - cot x)
sin x + sin y = 2 sin[l/2(x + y)] cos[l/2(x - y)]
sin x - sin y = 2 cos[l/2(x + y)] sin[l/2(x - y)]
cos x + cos y = 2 cos[l/2(x + y)] cos[l/2(x - y)]
cos x - cos y = - 2 sin[l/2(x + y)] sin[l/2(x - y)]
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(y - x)]/[sin x sin y]
sin2x - sin2y = cos2y - cos2x
= sin(x + y) sin(x - y)
cos2x - sin2y = cos2y - sin2x
= cos(x + y) cos(x - y)
sin(45" + x) = cos(45" - x), tan(45" + x) = cot(45" - x)
sin(45" - x) = cos(45" + x), tan(45" - x) = cot(45" + x)
Multlple and Half Angles
tan 2x = (2 tan x)/(l - tan2x)
cot 2x = (COt2X - 1)/(2 cot x)
sin(nx) = n sin x cosn-'x - (n) ,sin3x COS~X + (n)5sin5x cosW5x -
cos(nx) = cosnx - (n),sin2x cosw2x + (n),sin4x COS~X - . . .
(Note: (n)z, . . . are the binomial coefficients)
sin(x/2) = i J1/2o

COS(x/2) = f Jl2 (1 + cos x)
tan(x/2) = (sin x)/(l + cos x) = f J(1- COS x)/(I + COS x)

Three Angles Whose Sum = 180"
sin A + sin B + sin C = 4 cos(N2) cos(B/2) cos(CI2)
cos A + cos B + cos C = 4 sin(N2) sin(B/2) sin(C/2) + 1
sin A + sin B - sin C = 4 sin(N2) sin(B/2) cos(C/2)
cos A + cos B - cos C = 4 cos(N2) cos(B/2) sin(CI2) - 1
sin2A + sin2B + sin% = 2 cos A cos B cos C + 2
sinZA + sin2B - sin% = 2 sin A sin B cos C
tan A + tan B + tan C = tan A tan B tan C
cot(N2) + cOt(B/2) + cot(C/2) = cot(N2) cot(B/2) cot(C/2)
sin 2A + sin 28 + sin 2C = 4 sin A sin B sin C
sin 2A + sin 28 - sin 2C = 4 cos A cos B sin C