Summary of Properties of Trigonometric Functions:

  1. Reciprocal

cot x = 1/tan x OR tan x cot x = 1

sec x = 1/cos x OR cos x sec x = 1

csc x = 1/sin x OR sin x csc x = 1

  1. Quotient

Tan x = sin x/cos x = sec x/ csc x

Cot x = cos x/sin x = csc x / sec x

  1. Pythagorean

Cos2x + sin2x = 1

1 + tan2x = sec 2x

cot2x + 1 = csc2x

  1. Odd-Even

Sin(-x) = -sin x (odd)

Cos (-x) = cos x (even)

Tan (-x) = -tan x (odd)

Cot (-x) = -cot x (odd)

Sec (-x) = sec x (even)

Csc (-x) = -csc x (odd)

  1. Cofunction

Cos (90 - ) = sin ; cos(P/2 - ) = sin x

Cot(90-) = tan ; cot (P/2 x) = tan x

Csc(90=) = sec ; csc (P/2 x) = sec x

  1. Composite Argument

Cos (a-b) = cos a cos b +sin a sin b

Cos (a+b) = cos a cos b sin a sin b

Sin (a-b) = sin a cos b cos a sin b

Sin (a+b) = sin a cos b + cos a cos b

Tan (a-b) = (tan a tan b) / (1 + tan a tan b)

Tan (a=b) = (tan a + tan b) / (1 tan a tan b)

  1. Double Argument

Sin 2x = 2sin x cos x

Cos 2x = cos2x sin2x = 1 2 sin2x = 2 cos2x 1

Tan 2x = (2tan x) / (1-tan2x)

Cos2x = (1+cos2x)

Sin2x = (1-cos2x)

  1. Half Argument

Sin x = (1/2(1 cos x))

Cos x = (1/2(1+cos x)0

Tan x = (1/2(1-cos x) / (1 + cos x)

  1. Sum and Product

2 cos a cos b = cos (a+b) + cos (a-b)

2 sin a sin b = -cos (a+b) + cos (a-b)

2 sin a cos b = sin (a +b) + cos (a-b)

2 cos A sin B = sin (a+b) sin (a-b)

  1. Linear Combination of Sine and Cosine

A cos x + B sin x = C cos (x D); where

C = (A2 + B2), cos D = A/C, and sin D = B/C