Double and Half angle Identities

Notes:

Double angle and half angle identities

Identities for Double angles

  • Sin(2x) = sin(x + x) = sin(x) * cos(x) + cos(x) * sin(x) = 2sin(x)cos(x)

vvv Cosine has 3 outputs vvv

  • cos(2x) = cos(x + x) = cos(x) * cos(x) - sin(x) * sin(x) =
    cos2(x) - sin2(x)
    && (1 - sin2(x)) - sin2(x) = 1 - 2sin2(x)
    && Cos2(x) - (1 - cos2(x)) = 2cos2(x) - 1

  • tan(2x) = tan(x + x) = tan(x) + tan(x) / 1 - (tan(x))(tan(x)) = 2tan(x) / 1 - tan2(x)

Identities for Half angles

  • sin(x / 2) = +-√((1 - cos(x)) / 2)

  • cos(x / 2) = +-√((1 + cos(x)) / 2)

  • tan(x / 2) = +-√((1 - cos(x)) / (1 + cos(x)))

ALSO

  • tan(x / 2) = sin(x) / 1 + cos(x)