Trigonometric
Formulas and Relationships

Definition of Ratios
Geometric interpretation of Trig Functions
 Sine = Opposite / Hypotenuse     Cosine = Adjacent / Hypotenuse     Tangent = Opposite / Adjacent     Cosecant = Hypotenuse / Opposite     Secant = Hypotenuse / Adjacent     Cotangent = Adjacent / Opposite
 Fundamental Identities: sin x / cos x = tan x cos x / sin x = cot x = 1 / tan x sec x = 1 / cos x csc x = 1 / sin x sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x = 1 / cos2 x cot2 x + 1 = csc2 x = 1 / sin2 x
 Reduction Formulas: sin (90 - x) = cos x cos (90 - x) = sin x tan (90 - x) = 1 / tan x = cot x More Reduction Formulas

Sum Identities:
sin (a + b) = sin a cos b + sin b cos a
cos (a + b) = cos a cos b - sin b sin a
tan (a + b) = (tan a + tan b) / (1 - tan a tan b)

Difference Identities:
sin (a - b) = sin a cos b - sin b cos a
cos (a - b) = cos a cos b + sin b sin a
tan (a - b) = (tan a - tan b) / (1 + tan a tan b)

Double Angle Relationships:
sin2 x = (1 - cos 2x) / 2
cos2 x = (1 + cos 2x) / 2
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)

Half Angle Relationships:
sin (x/2) = [(1 - cos x) / 2]½
cos (x/2) = [(1 + cos x) / 2]½
tan (x/2) = [(1 - cos x) / (1 + cos x)]½
= (1 - cos x) / sin x = sin x / (1 + cos x)

Factorization Equations:
sin a + sin b = 2 sin[(a + b)/2] cos[(a - b)/2]
sin a - sin b = 2 cos[(a + b)/2] sin[(a - b)/2]
cos a + cos b = 2 cos[(a + b)/2] cos[(a - b)/2]
cos a - cos b = -2 sin[(a + b)/2] sin[(a - b)/2]

Product Formulas:
sin x sin y = [cos(x - y) - cos(x + y)] / 2
sin x cos y = [sin(x - y) + sin(x + y)] / 2
cos x cos y = [cos(x - y) + cos(x + y)] / 2
 Law of Sines: a /sin A = b /sin B = c /sin C Law of Cosines: a2 = b2 + c2 - 2bc cos A If A = 90 degrees, then cos 90 = 0, and the equation reduces to the Pythagorean Theorem : a2 = b2 + c2

Pythagorean Triples:
x and y are positive integers, x > y :
Let a = x2 + y2 , b = 2xy , and c = x2 - y2
Then a2 = b2 + c2

a sin x + b cos x:
Let q = arctan (- b / a)
Then a [sin x - (- b cos x / a)]
= a (sin x - tan q cos x )
= a (sin x - sin q cos x / cos q)
= a (sin x cos q - cos x sin q) / cos q
= a sin (x - q) / cos q