Trigonometric Identities

Cos[x]^2 + Sin[x]^2 = 1

Tan[x] = Sin[x] / Cos[x]

Cos[-x] = Cos[x]

Sin[-x] = -Sin[x]

Tan[-x] = -Tan[x]

Cos[x + 2 Pi n] = Cos[x]

Sin[x + 2 Pi n] = Sin[x]

Trigonometric functions of Inverse Trigonometric functions

Cos[ArcSin[x]] = Sqrt[1 - x^2]

Sin[ArcCos[x]] = Sqrt[1 - x^2]

Sec[ArcTan[x]] = Sqrt[1 + x^2]

Csc[ArcCot[x]] = Sqrt[1 + x^2]

Inverse Trigonometric functions of Trigonometric functions

ArcCos[Sin[x]] = x - Pi/2

ArcTan[Cot[x]] = x - Pi/2

Angle Addition or Subtraction

Cos[x +/- y] = Cos[x] Cos[y] -/+ Sin[x] Sin[y]

Sin[x +/- y] = Sin[x] Cos[y] +/- Cos[x] Sin[y]

Tan[x +/- y] = (Tan[x] +/- Tan[y]) / (1 -/+ Tan[x] Tan[y])

Tan[x +/- y] = (Cot[x] +/- Cot[y]) / (Cot[x] Cot[y] -/+ 1)

Angle Addition leads to multiple angle identities

Cos[2 x] = Cos[x]^2 - Sin[x]^2 = 2 Cos[x]^2 - 1 = 1 - 2 Sin[x]^2

Sin[2 x] = 2 Sin[x] Cos[x]

Tan[2 x] = (2 Tan[x]) / (1 - Tan[x]^2)

Tan[2 x] = (2 Cot[x]) / (Cot[x]^2 - 1)

Cos[3 x] = 4 Cos[x]^3 - 3 Cos[x]

Angle Addition also leads to half angle identities

Cos[x/2] = Sqrt[(1 + Cos[x])/2]

Angle Addition also leads to reducing 2 funciotn to one trigonometric function

A Cos[x] + B Sin[x] = Sqrt[A^2 + B^2] Cos[y - x], where Tan[y] = A/B

Complex Triginometric Identities

Exp[+/-I x] = Cos[x] +/- I Sin[x]

Cos[x] = (Exp[I x] + Exp[-I x]) / 2

Sin[x] = (Exp[I x] - Exp[-I x]) / (2 I)

Tan[x] = (Exp[I x] - Exp[-I x]) / I (Exp[I x] + Exp[-I x])

Exp[+/-I x]^n = Exp[+/-I x n] = Cos[n x] +/- I Sin[n x]

Logarithmic identities

Log[x y] = Log[x] + Log[y]

Log[x1 x2 x3] = Log[x1] + Log[x2] + Log[x3]

Log[x^a] = a Log[x]

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last updated 03-apr-2005