Trigonometric function

  1. Basic formula

    sin2 α + cos2 α + = 1 tan α * cottan α = 1 tan α = sinαcosα = 1cottanα cottan α = cosαsinα = 1tanα 1+ tan2 α = 1 cos2 α = sec2 α 1+ cottan2 α = 1 sin2 α = cossec2 α

  2. Addition formula

    sin(α±β) = sinαcosβ±cosαsinβ cos(α±β) = cosαcosβ±sinαsinβ tan(α±β) = cotαcotβ±1 cotβ±cotα tan(α±β) = tanα±tanβ 1±tanαtanβ

  3. The sum of trigonometric function

    cosα+cosβ = 2cos α+β 2 cos α-β 2 cosα-cosβ = -2sin α+β 2 sin α-β 2 sinα+sinβ = 2sin α+β 2 cos α-β 2 sinα-sinβ = 2cos α+β 2 sin α-β 2 sinα+cosα = 2sin(α+π4) = 2cos(π4-α) sinα-cosα = 2sin(α-π4) = -2cos(π4-α) tanα+tanβ = sin(α+β) cosαcosβ tanα-tanβ = sin(α-β) cosαcosβ cottanα+cottanβ = sin(α-β) sinαsinβ cottanα-cottanβ = sin(β-α) sinαsinβ tanα+cottanα = 2cossec2α tanα-cottanα = -2cossec2α

  4. The product of trigonometric function

    cosαcosβ = 12 [ cos(α-β) + cos(α+β) ] sinαsinβ = 12 [ cos(α-β) - cos(α+β) ] sinαcosβ = 12 [ sin(α+β) + sin(α-β) ] tanαtanβ = tanα+tanβ cottanα+cottanβ = - tanα-tanβ cottanα+cottanβ cottanαcottanβ = cottanα+cottanβ tanα+tanβ = - cottanα-cottanβ tanα-tanβ cottanαtanβ = cottanα+tanβ tanα+cottanβ = - cottanα-tanβ tanα-cottanβ

  5. The power of trigonometric function

    sin2α = 12 (1-cos2α) cos2α = 12 (1+cos2α) tan2α = 1-cos2α 1+cos2α sin3α = 14 (3sinα-sin3α) cos3α = 14 (3cosα+cos3α)

  6. Double angle formula

    sin2 α = 2sinαcosα cos2 α = 2 cos2α - 1 = 1 - 2 sin2α = cos2α - sin2α tan2 α = 2tanα 1-tan2α cottan2 α = cottan2α-1 2cottanα = cottanα-tanα 2 sin3α = 3sinα - 4sin3α cos3α = 4cos3α - 3cosα tan3α = 3tanα-tan3α 1-3tan2α cottan3α = cottan3α-3cottanα 3 cottan2α-1

  7. Half angle formula

    sinα2 = ± 1-cosα2 cosα2 = ± 1+cosα2 tanα2 = sinα1+cosα = 1-cosαsinα = ± 1-cosα1+cosα cottanα2 = sinα1-cosα = 1+cosαsinα = ± 1+cosα1-cosα sinα = 2tanα2 1+tan2α2 cosα = 1-tan2α2 1+tan2α2 tanα = 2tanα2 1-tan2α2 |cosα±sinα| = 1+sin2α 1+cosα = 2cos2α2 1-cosα = 2sin2α2 1+sinα = (sinα2+cosα2)2 = 2cos2 (π4-α2) 1-sinα = (sinα2-cosα2)2 = 2sin2 (π4-α2)

  8. Interior angle formula

    α+β+γ = 180° sinα+sinβ+sinγ = 4cosα2cosβ2cosγ2 cosα+cosβ+cosγ = 4sinα2 sinβ2 sinγ2 +1 sinα+sinβ-cosγ = 4sinα2 sinβ2 cosγ2 cosα+cosβ-cosγ = 4cosα2 cosβ2 sinγ2 -1 sin2α+sin2β+sin2γ = 2cosαcosβcosγ +2 sin2α+sin2β-sin2γ = 2sinαsinβsinγ sin2α+sin2β+sin2γ = 4sinαsinβsinγ sin2α+sin2β-sin2γ = 4cosαcosβsinγ tanα+tanβ+tanγ = tanαtanβtanγ cottanα2+cottanβ2+cottanγ2 = cottanα2cottanβ2cottanγ2 cottanαcottanβ+cottan βcottanγ+cottanα cottanγ = 1