A not-quite-exhaustive, but certainly extensive list of trig formulas and identities. Almost everything you might need is here, along with some things you may not have seen yet.
Basic Definitions
This is a list of commonly used trig identities. Everything a beginning to intermediate student would need is here. While an exhaustive list might be helpful, all other identities can be derived from these. Even most of the following can be derived from each other.
Quotient Identities:
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)
Reciprocal Functions:
sin(x) = 1/csc(x)
cos(x) = 1/sec(x)
tan(x) = 1/cot(x)
csc(x) = 1/sin(x)
sec(x) = 1/cos(x)
cot(x) = 1/tan(x)
Pythagorean Identities:
sin2(x) + cos2(x) = 1
1+tan2(x) = sec2(x)
cot2(x) + 1 = csc2(x)
Inverse Functions:
Remember that trig inverses are only partial inverse functions, and don’t cover the entire number line.
sin-1(x) = arcsin(x)
cos-1(x) = arccos(x)
tan-1(x) = arctan(x)
csc-1(x) = arccsc(x)
sec-1(x) = arcsec(x)
cot-1(x) = arccot(x)
Co-Functions, Even-Odd and Sum-Difference Formulas
Co-Function Identities:
(here, p = pi radians = 3.14159 radians = 180 degrees)
sin(p/2 - x) = cos(x)
cos(p/2 - x) = sin(x)
tan(p/2 - x) = cot(x)
csc(p/2 - x) = sec(x)
sec(p/2 - x) = csc(x)
cot(p/2 - x) = tan(x)
Even-Odd Identities
sin(-x) = -sin(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)
csc(-x) = -csc(x)
sec(-x) = sec(x)
cot(-x) = -cot(x)
Sum-Difference Formulas
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)]
Sum-Difference Formulas for Inverses:
arcsin(x) + arcsin(y) = arcsin[x*sqrt(1-y2) + y*sqrt(1-x2)]
arcsin(x) - arcsin(y) = arcsin[x*sqrt(1-y2) - y*sqrt(1-x2)]
arccos(x) + arccos(y) = arccos(xy - sqrt[(1-x2)*(1-y2)]
arccos(x) - arccos(y) = arccos(xy + sqrt[(1-x2)*(1-y2)]
arctan(x) + arctan(y) = arctan([x+y] / [1-xy])
arctan(x) - arctan(y) = arctan([x-y] / [1+xy])
Tangent of an Average:
tan([x+y]/2) = [sinx+siny]/[cosx+cosy]
tan([x+y]/2) = - ([cosx-cosy] / [sin(x)-sin(y)])
Double Angle, Power-Reduction and Half Angle Formulas
Double Angle Formulas:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos2(x) -sin2(x)
cos(2x) = 2cos2(x)-1
cos(2x) = 1-2sin2(x)
tan(2x) = [2tan(x)] / [1-tan2(x)]
Power-Reduction Formulas:
sin2(x) = (1/2)*[1-cos(2x)]
cos2(x) = (1/2)*[1+cos(2x)]
tan2(x) = [1-cos(2x)] / [1+cos(2x)]
Half Angle Formulas:
sin(x/2) = ±sqrt [(1/2)*(1-cos(x))]
cos(x/2) = ±sqrt [(1/2)*(1+cos(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)
Sum-to-Product and Product-to-Sum Formulas
Sum-to-Product Formulas:
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)
Product-to-Sum Formulas:
sin(x)sin(y) = (1/2)*[cos(x-y) - cos(x+y)]
cos(x)cos(y) = (1/2)*[cos(x-y) + cos(x+y)]
sin(x)cos(y) = (1/2)*[sin(x+y) + sin(x+y)]
cos(x)sin(y) = (1/2)*[sin(x-y) - sin(x+y)]
Inverse Functions and Compositions
Inverse Function Relationships:
(remember that p = pi radians)
arcsin(x) + arccos(x) = p/2
arctan(x) + arccot(x) = p/2
arctan(x) + arctan(1/x) = p/2, if x>0
arctan(x) + arctan(1/x) = -p2, if x<0
Compositions of Trig Functions and Inverses:
sin(arccos(x)) = sqrt[1-x2]
tan(arcsin(x)) = x/ sqrt[1-x2]
sin(arctan(x)) = x/ sqrt[1+x2]
tan(arccos(x)) = sqrt[1-x2] / x
cos(arctan(x)) = 1/ sqrt[1+x2]
cot(arcsin(x)) = sqrt[1-x2] / x
cos(arcsin(x)) = sqrt[1-x2]
cot(arccos(x)) = x/ sqrt[1-x2]
The Exponential Function and the Trig Functions
The following definitions relate the trig functions you already know with the exponential function e and the natural logarithm function ln. This list assumes prior knowledge of the e and ln functions, and their relationships.
The imaginary number i is also used in this section. For review: i = sqrt(-1)
Exponential Definitions of Trig Functions
sin(x) = [eix - e-ix] / [2i]
cos(x) = [eix + e-ix] / [2i]
tan(x) = [eix - e-ix] / [i*(eix + e-ix)]
csc(x) = [2i] / [eix - e-ix]
sec(x) = [2i] / [eix + e-ix]
cot(x) = [I*(eix + e-ix)] / [eix - e-ix]
Exponential Definitions of Inverse Trig Functions
arcsin(x) = -i*ln(ix+sqrt[1-x2])
arccos(x) = -i*ln(x+sqrt[x2-1])
arctan(x) = (i/2)*ln([i+x]/[i-x])
arccsc(x) = -i*ln( (i/x) + sqrt[1-(1/x2)] )
arcsec(x) = -i*ln( (1/x) + sqrt[1-(i/x2)] )
arccot(x) = (i/2)*ln([x-i]/[x+i])
Trig Definition of the Exponential Function and Natural Log
cis(x) = eix = cos(x) + i*sin(x)
arccis(x) = (1/i)*ln(x)
Note: The cis(x) notation is not used by all teachers/professors, so check with yours about the best notation.
This post is part of the series: Trig Help
Everything you need to know to get through your trig class. Whether in high school or college, the tips, tricks, formulas and methods you need can all be found here.
