和差公式:

sin(α+β)=sinαcosβ+cosαsinβ sin(\alpha + \beta) = sin\alpha cos\beta + cos\alpha sin\beta
sin(α−β)=sinαcosβ−cosαsinβ sin(\alpha - \beta) = sin\alpha cos\beta - cos\alpha sin\beta
cos(α+β)=cosαcosβ−sinαsinβ cos(\alpha + \beta) = cos\alpha cos\beta - sin\alpha sin\beta
cos(α−β)=cosαcosβ+sinαsinβ cos(\alpha - \beta) = cos\alpha cos\beta + sin\alpha sin\beta

tan(α+β)=tanα+tanβ1−tanαtanβ tan(\alpha + \beta) = \frac{tan\alpha+tan\beta}{1-tan\alpha tan\beta}

tan(α−β)=tanα−tanβ1+tanαtanβ tan(\alpha - \beta) = \frac{tan\alpha-tan\beta}{1+tan\alpha tan\beta}

cot(α+β)=cotαcotβ−1cotβ+cotα cot(\alpha + \beta) = \frac{cot\alpha cot\beta - 1}{cot\beta + cot\alpha}

cot(α−β)=cotαcotβ+1cotβ−cotα cot(\alpha - \beta) = \frac{cot\alpha cot\beta + 1}{cot\beta - cot\alpha}

和差化积公式:

sinα+sinβ=2sinα+β2cosα−β2 sin\alpha + sin\beta = 2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}
sinα−sinβ=2cosα+β2sinα−β2 sin\alpha - sin\beta = 2cos\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}
cosα+cosβ=2cosα+β2cosα−β2 cos\alpha + cos\beta = 2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}
cosα−cosβ=−2sinα+β2sinα−β2 cos\alpha - cos\beta = -2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}

口诀：
正加正，正在前；
正减正，余在前；
余加余，余并肩；
余减余，负正弦．

积化和差公式:

sinαcosβ=12[sin(α+β)+sin(α−β)] sin{\alpha} cos{\beta} = \frac{1}{2}[{sin(\alpha+\beta) + sin(\alpha-\beta)}]

sin(α+β)=sinαcosβ+cosαsinβ sin(\alpha+\beta) = sin\alpha cos\beta + cos\alpha sin\beta

cos(α+β)=cosαcosβ−sinαsinβ cos(\alpha + \beta) = cos\alpha cos\beta - sin\alpha sin\beta

tan(α+β)=tanα+tanβ1−tanαtanβ tan(\alpha + \beta) = \frac{tan\alpha + tan\beta}{1 - tan\alpha tan\beta}

tan(α−β)=tanα−tanβ1+tanαtanβ tan(\alpha - \beta) = \frac{tan\alpha - tan\beta}{1 + tan\alpha tan\beta}

cot(α+β)=cotαcotβ−1cotβ+cotα cot(\alpha + \beta) = \frac{cot\alpha cot\beta - 1}{cot\beta + cot\alpha}

cot(α−β)=cotαcotβ+1cotβ−cotα cot(\alpha - \beta) = \frac{cot\alpha cot\beta + 1}{cot\beta - cot\alpha}

帮助记忆:
口诀：
正余余正，正加正减；
余余正正，余加负余减；

倍角公式:

sin2α=2sinαcosα sin2\alpha = 2sin\alpha cos\alpha

cos2α=cos2α−sin2α=1−2sin2α=2cos2α−1 cos2\alpha = cos^2{\alpha} - sin^2{\alpha} = 1-2sin^2{\alpha}=2cos^2{\alpha}-1 ★★

tan2α=2tanα1−tan2α tan2\alpha = \frac{2tan\alpha}{1-tan^2{\alpha}}

cot2α=cot2α−12cotα cot2\alpha = \frac{cot^2{\alpha}-1}{2cot\alpha}

半角公式

sin2α2=1−cosα2 sin^2\frac{\alpha}{2}=\frac{1-cos\alpha}{2}

cos2α2=1+cosα2 cos^2\frac{\alpha}{2}=\frac{1+cos\alpha}{2}

tanα2=sinα1+cosα=1−cosαsinα=±1−cosα1+cosα tan\frac{\alpha}{2} = \frac{sin\alpha}{1+cos\alpha}=\frac{1-cos\alpha}{sin\alpha}=\pm \sqrt{\frac{1-cos\alpha}{1+cos\alpha}}

cotα2=1+cosαsinα=sinα1−cosα=±1+cosα1−cosα cot\frac{\alpha}{2} = \frac{1+cos\alpha}{sin\alpha}=\frac{sin\alpha}{1-cos\alpha}=\pm \sqrt{\frac{1+cos\alpha}{1-cos\alpha}}

（正负由α2\frac{\alpha}{2}所在的象限决定）

万能公式

sinα=2tanα21+tan2α2 sin\alpha=\frac{2tan{\frac{\alpha}{2}}}{1+tan{^2}{\frac{\alpha}{2}}}

cosα=1−tan2α21+tan2α2 cos\alpha=\frac{1-tan{^2}{\frac{\alpha}{2}}}{1+tan{^2}{\frac{\alpha}{2}}}

tanα=2tanα21−tan2α2 tan\alpha = \frac{2tan{\frac{\alpha}{2}}}{1-tan{^2}{\frac{\alpha}{2}}}

辅助角公式

asinα+bcosα=a2+b2sin(α+φ),tanφ=ba asin\alpha + bcos\alpha = \sqrt{a^2+b^2}sin(\alpha+\varphi), tan\varphi=\frac{b}{a}