# Half Angle Formulas

Updated 2009-01-22 12:04

## All identities below hold for right triangles

x is the angle between the adjacent side and the hypotenuse.

## Basic Relationships (left side = right side)

sin x(opposite side)/hypotenuse
csc xhypotenuse/(opposite side)

sin2x + cos2x1
tan x(sin x)/(cos x)
csc x1/(sin x)
sec x1/(cos x)
cot x(cos x)/(sin x)

Updated: use the new 'combine' link on the right side of this page to combine tables. pools the above 2 tables

## Each in terms of the others (left side = right side)

cos xsqrt(1 - sin2x)
tan x(sin x)/sqrt(1 - sin2x)
csc x1/(sin x)
sec x1/sqrt(1 - sin2x)
cot xsqrt(1 - sin2x)/(sin x)

sin xsqrt(1 - cos2x)
tan xsqrt(1 - cos2x)/(cos x)
csc x1/sqrt(1 - cos2x)
sec x1/(cos x)
cot x(cos x)/sqrt(1 - cos2x)

sin x(tan x)/sqrt(1 + tan2x)
cos x1/sqrt(1 + tan2x)
csc xsqrt(1 + tan2x)/(tan x)
sec xsqrt(1 + tan2x)
cot x1/(tan x)

sin x1/(csc x)
cos xsqrt(csc2 - 1)/(csc x)
tan x1/sqrt(csc2x - 1)
sec x(csc x)/sqrt(csc2x - 1)
cot xsqrt(csc2x - 1)

sin xsqrt(sec2x - 1)/(sec x)
cos x1/(sec x)
tan xsqrt(sec2x - 1)
csc x(sec x)/sqrt(sec2x - 1)
cot x1/sqrt(sec2x - 1)

sin x1/sqrt(1 + cot2x)
cos x(cot x)/sqrt(1 + cot2x)
tan x1/(cot x)
csc xsqrt(1 + cot2x)
sec xsqrt(1 + cot2x)/(cot x)

Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 6 tables

## Double-Angle Formulas (left side = right side)

sin(2x)2 sin x cos x
sin(2x)(2 tan x)/(1 + tan2x)
cos(2x)cos2x - sin2x
cos(2x)2 cos2 x - 1
cos(2x)1 - 2 sin2x
cos(2x)(1 - tan2x)/(1 + tan2x)
tan(2x)(2 tan x)/(1 - tan2x)
cot(2x)(cot2x - 1)/(2 cot x)

## Triple-Angle Formulas (left side = right side)

sin(3x)3 sin x - 4 sin3x
cos(3x)4 cos3x - 3 cos x
tan(3x)(3 tan x - tan3x)/(1 - 3 tan2x)
cot(3x)(3 cot x - cot3x)/(1 - 3 cot2x)

## Half-Angle Formulas (left side = right side)

sin(x/2)±sqrt[(1 - cos x)/2]
cos(x/2)±sqrt[(1 + cos x)/2]
tan(x/2)csc x - cot x
tan(x/2)±sqrt[(1 - cos x)/(1 + cos x)]
tan(x/2)(sin x)/(1 + cos x)
tan(x/2)(1 - cos x)/(sin x)
cot(x/2)csc x + cot x
cot(x/2)±sqrt[(1 + cos x)/(1 - cos x)]
cot(x/2)(sin x)/(1 - cos x)
cot(x/2)(1 + cos x)/(sin x)

Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 3 tables

## Angle-Addition Formulas (left side = right side)

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)/(1 - tan x tan y)
tan(x-y)(tan x - tan y)/(1 + tan x tan y)
tan[(x+y)/2](sin x + sin y)/(cos x + cos y)
tan[(x+y)/2]-(cos x - cos y)/(sin x - sin y)

## Product-to-Sum Formulas (left side = right side)

cos x cos y[cos(x-y) + cos(x+y)]/2
sin x sin y[cos(x-y) - cos(x+y)]/2
sin x cos y[sin(x+y) + sin(x-y)]/2
cos x sin y[sin(x+y) - sin(x-y)]/2

## Sum-to-Product Formulas (left side = right side)

sin x + sin y2 sin[(x+y)/2] cos[(x-y)/2]
cos x + cos y2 cos[(x+y)/2] cos[(x-y)/2]
cos x - cos y-2 sin[(x+y)/2] sin[(x-y)/2]
sin x - sin y2 cos[(x+y)/2] sin[(x-y)/2]

Updated: use the new 'combine' link on the right side of this page to combine tables. pools the last 3 tables

## Miscellaneous Formulas (left side = right side)

cos x + i sin xeix
cos(-x) + i sin(-x)e-ix
cos x - i sin xe-ix
-1i2
-1e
sin x(eix - e-ix)/(2i)
cos x(eix + e-ix)/(2)
tan x(eix - e-ix)/[i(eix + e-ix)]
cos(nx) + i sin(nx)(cos x + i sin x)n
tan[(n+1)x][tan(nx) + tan(x)]/[1 - tan(nx) tan(x)]
cot[(n+1)x][cot(nx) cot(x) - 1]/[cot(nx) + cot(x)]

Reference: http://en.wikipedia.org/wiki/Trigonometric_identity