Table of Trigonometric Functions

Trigonometry came from the Greek words ‘trigon-triangle’ and ‘metron-measure’. Trigonometric functions generally define the function of angles. Relating the angles of triangle to its length of sides is the common use of trigonometric function. Table of trigonometric functions are generally applied in modeling periodic phenomena and study of triangles. Sin, cos, and tan are the most memorable trigonometric functions. Table of trigonometric functions refers to tabulating the trigonometric functions and trigonometric identities in order.

 

Formulas in the Table of Trigonometric Functions:

 

  • Sum-Difference Formula:

            Sin(A + B) = SinACosB + CosASinB                                   Sin(A − B) = SinACosB − CosASinB

            Cos(A + B) = CosACosB − SinASinB                                  Cos(A − B) = CosACosB + SinASinB

            Tan(A + B) = `(TanA + TanB)/ (1 - TanATanB)`                                        Tan(A − B) = `(TanA - TanB)/ (1 + TanATanB)`

  • Sum-to-Product Formula:

            SinA + SinB = 2Sin(`(A + B) / 2` ) Cos(`(A - B) / 2` )                SinA − SinB = 2Cos(`(A + B) / 2` ) Sin(`(A - B) / 2` )

            CosA − CosB = 2Cos(`(A + B) / 2` ) Cos(`(A - B) / 2` )            CosA − CosB = −2Sin(`(A + B) / 2` ) Sin(`(A - B) / 2` )

  • Product-to-Sum Formula:

            SinASinB = `1/2` [Cos(A − B) − Cos(A + B)]                           CosACosB = `1/2` [Cos(A − B) + Cos(A + B)]

            SinACosB = `1/2` [Sin(A + B) + Sin(A − B)]                            CosASinB = `1/2` [Sin(A + B) − Sin(A − B)]

 

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Identities in the Table of Trigonometric Functions:

 

  • Reciprocal identity,

              sin x = `1 / csc x` ,                  csc 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 identity,

              tan x = `sinx / cosx` ,

              cot x = `cosx / sinx` .

  • Pythagorean identity,

              sin2x + cos2x = 1,

              1 + tan2x = sec2x,

              1 + cot2x = csc2x.

  • Co-function identity,

              sin (90° − x) = cos x,            cos (90° – x) = sin x,               tan (90° − x) = cot x,

              csc (90° – x) = sec x,            sec (90° − x) = csc x,              cot (90° − x) = tan x.

  • Even-Odd identity,

              sin (−x) = −sin x,                  cos (−x) = cos x,                     tan (−x) = −tan x,

              csc (−x) = −csc x,                 sec (−x) = sec x,                     cot (−x) = −cot x.

 

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Example Using Table of Trigonometric Functions:

 

Ex: Prove that, sin4x + 2sin2x cos2x + cos4x = tan x cot x

Proof:     L.H.S. = sin4x + 2sin2x cos2x + cos4x

                               => (sin2x)2 + 2sin2x cos2x + (cos2x)2

                               => (sin2x + cos2x)2

                               => (1)2

                               => 1

                   R.H.S. = tan x cot x

                               => `(sinx / cosx) xx (cosx / sinx )`

                               => 1

              => L.H.S. = R.H.S.

          Hence Proved that, sin4x + 2sin2x cos2x + cos4x = tanx cotx.