Trigonometric Formulas

Relations Between Trigonometric Functions

cosX = 1 / sinX

sinX = 1 / cscX

secX = 1 / cosX

cosX = 1 / secX

tanX = 1 / cotX

cotX = 1 / tanX

tanX = sinX / cosX

cotX = cosX / sinX



Pythagorean Identities

sin 2X + cos 2X = 1

1 + tan 2X = sec 2X

1 + cot 2X = csc 2X


Negative Angle Identities

sin(-X) = - sinX , odd function

csc(-X) = - cscX , odd function

cos(-X) = cosX , even function

sec(-X) = secX , even function

tan(-X) = - tanX , odd function

cot(-X) = - cotX , odd function


Cofunctions Identities

sin(pi/2 - X) = cosX

cos(pi/2 - X) = sinX

tan(pi/2 - X) = cotX

cot(pi/2 - X) = tanX

sec(pi/2 - X) = cscX

csc(pi/2 - X) = secX


Addition Formulas

cos(X + Y) = cosX cosY - sinX sinY

cos(X - Y) = cosX cosY + sinX sinY

sin(X + Y) = sinX cosY + cosX sinY

sin(X - Y) = sinX cosY - cosX sinY

tan(X + Y) = [ tanX + tanY ] / [ 1 - tanX tanY]

tan(X - Y) = [ tanX - tanY ] / [ 1 + tanX tanY]

cot(X + Y) = [ cotX cotY - 1 ] / [ cotX + cotY]

cot(X - Y) = [ cotX cotY + 1 ] / [ cotX - cotY]



Sum to Product Formulas

cosX + cosY = 2cos[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]

sinX + sinY = 2sin[ (X + Y) / 2 ] cos[ (X - Y) / 2 ]
Difference to Product Formulas

cosX - cosY = - 2sin[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]

sinX - sinY = 2cos[ (X + Y) / 2 ] sin[ (X - Y) / 2 ]

Product to Sum/Difference Formulas

cosX cosY = (1/2) [ cos (X - Y) + cos (X + Y) ]

sinX cosY = (1/2) [ sin (X + Y) + sin (X - Y) ]

cosX sinY = (1/2) [ sin (X + Y) - sin[ (X - Y) ]

sinX sinY = (1/2) [ cos (X - Y) - cos (X + Y) ]

Difference of Squares Formulas

sin 2X - sin 2Y = sin(X + Y)sin(X - Y)

cos 2X - cos 2Y = - sin(X + Y)sin(X - Y)

cos 2X - sin 2Y = cos(X + Y)cos(X - Y)

Double Angle Formulas

sin(2X) = 2 sinX cosX

cos(2X) = 1 - 2sin 2X = 2cos 2X - 1

tan(2X) = 2tanX / [ 1 - tan 2X ]

Multiple Angle Formulas

sin(3X) = 3sinX - 4sin 3X

cos(3X) = 4cos 3X - 3cosX

sin(4X) = 4sinXcosX - 8sin 3XcosX

cos(4X) = 8cos 4X - 8cos 2X + 1

Half Angle Formulas

sin (X/2) = + or - SQRT [ (1 - cosX) / 2 ]

cos (X/2) = + or - SQRT [ (1 + cosX) / 2 ]

tan (X/2) = + or - SQRT [ (1 - cosX) / (1 + cosX) ]

= sinX / (1 + cosX) = (1 - cosX) / sinX

Power Reducing Formulas

sin 2X = 1/2 - (1/2)cos(2X))

cos 2X = 1/2 + (1/2)cos(2X))

sin 3X = (3/4)sinX - (1/4)sin(3X)

cos 3X = (3/4)cosX + (1/4)cos(3X)

sin 4X = (3/8) - (1/2)cos(2X) + (1/8)cos(4X)

cos 4X = (3/8) + (1/2)cos(2X) + (1/8)cos(4X)

sin 5X = (5/8)sinX - (5/16)sin(3X) + (1/16)sin(5X)

cos 5X = (5/8)cosX + (5/16)cos(3X) + (1/16)cos(5X)

sin 6X = 5/16 - (15/32)cos(2X) + (6/32)cos(4X) - (1/32)cos(6X)

cos 6X = 5/16 + (15/32)cos(2X) + (6/32)cos(4X) + (1/32)cos(6X)

Trigonometric Functions Periodicity

sin (X + 2Pi) = sin X , period 2Pi

cos (X + 2Pi) = cos X , period 2Pi

sec (X + 2Pi) = sec X , period 2Pi

csc (X + 2Pi) = csc X , period 2Pi

tan (X + Pi) = tan X , period Pi

cot (X + Pi) = cot X , period Pi