# Trigonometry Formulas, Functions and Identities – Free Homework Tutor Help, Videos and pdf

Things to Remember: Important Formula for Trigonometry

In mathematics, trigonometric identities are equalities involving trigonometric functions that are true for all values of the occurring variables. These identities are useful whenever expressions involving trigonometric functions need to be simplified.

The step involves, first using the substitution rule with a
trigonometric function, and then simplifying the resulting integral using with a trigonometric identity when the integration is given in a non-trigonometric identities.

Important List of Trigonometric Formulas:

 Notation of Trigonometric Functions : The Six trigonometric function notations are Sine Function : sin (theta) Cosine Function :cos (theta) Tangent Function : tan (theta) Cotangent Function : cot (theta) Secant Function : sec (theta) Cosecant Function : cosec (theta) or csc (theta) Bacis Trigonometric Formula: In a right Triangle ABC, B is a right angle, then sin() = Perpendicular / Hypotenuse cos() = Base / Hypotenuse tan() = Perpendicular / Base cot() = Base / Perpendicular sec() = Hypotenuse / Base cosec() or csc() = Base/Perpendicular Reciprocal Identities : sin x = 1/cosec x , cosec x = 1/sin x cos x = 1/sec x , sec x = 1/cos x tan x = 1/cot x , cot x = 1/tan x Quotient Identities : tan x = 1/cot x cot x = 1/tan x Pythagorean Identities :The Pythagorean identities are follow the concept of Pythagorean theorem i.e. (Hypotenuse)2 = (Base)2 + (Perpendicular)2. This simply known as Pythagorean trigonometric identity. we have three identities – sin 2x + cos2x = 1 tan2x + 1 = cot2x [we get this identity by dividing first by cos 2x] 1 + cot2x = cosec2x or csc2x [we get this identity by dividing first by sin 2x] Addition and Subtraction Formulas : These are also known as Angle Sum and Difference Identities. These are 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) Double Angle Formulas : When we substituting x = y in the sum of addition formula. we get these double angle formula sin (x + x) = sin x cos x + cos x sin x sin 2x = 2 sin x cos y cos (x + x) = cos x cos x – sin x sin x cos 2x = cos2 x – sin 2 x cos 2x = cos2x – (1 – cos 2 x) = 2 cos2 x – 1 [cos 2x in terms of cosine function] cos 2x = 1 – sin2 x – sin 2 x = 1 – 2 sin 2 x [cos 2x in terms of sine function] cos 2x = (1 – tan 2 x) / (1 + tan2 x) [cos 2x in terms of tangent function] tan (x + x) = (tan x + tan x) /(1 – tan x tan x) tan 2x = 2 tan x /(1 – tan2x) Triple Angle Formula : sin 3x = 3 sin x – 4 sin3x cos 3x = 4cos3x – 3 cos x tan 3x = (3 tan x – tan3 x) / (1 – 3 tan 2 x) Power Reduction Formulas : The sin2 x, cos2 x in terms of double angle formula sin2 x = (1 – cos 2x)/2 cos2 x = (1 + cos 2x)/2 tan2 x = (1 – cos 2x)/(1 + cos 2x) sin2 x cos2 x = (1 – cos 2x)(1 + cos 2x)/4 = (1 – cos2 2x)/4 = (1 – cos 4x)/8 Half Angle Formulas : When we substitute x/2 in place of x in power reduction formula and solve for sin x/2 and cos x/2, we get cos x/2 = √(1 + cos x)/2 sin x/2 = √(1 – cos x)/2 tan x/2 = sin x / (1 + cos x) = (1 – cos x) / sin x Product to Sum Identities: : When we expanding the addition formula in RHS, we get 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 Identities: : When we substituting x by (x + y)/2 and y by (x – y)/2 in the product to sum formula in RHS, we get cos x + cos y = 2 cos (x + y)/2 cos (x – y)/2 sin x + sin y = 2 sin (x + y)/2 cos (x – y)/2 cos x – cos y = – sin (x + y)/2 sin (x – y)/2 sin x – sin y = 2 cos (x + y)/2 sin (x – y)/2 Inverse Trigonometric Functions : Every trigonometric function can be related directly to every other trigonometric function and these relations can be expressed by means of inverse trigonometric functions, we have sin -1 x + cos -1 y = /2 tan-1 x + cot -1 y = /2 If x > 0, tan-1 x + tan -1 1/x = /2and if x < 0, tan-1 x + tan -1 1/x = – /2 sin (cos-1 x) = cos (sin-1 x) = √(1 – x2) sin (tan-1 x) = x/√(1 + x2) cos (tan-1 x) = 1/√(1 + x2) tan (sin-1 x) = x/√(1 – x2) tan (cos-1 x) = √(1 – x2) / x