三角函数公式总结

基础

一般公式

$sin(\alpha)+cos(\alpha)=1$
$tan(\alpha)=\frac{sin(\alpha)}{cos(\alpha)}$

诱导公式

1.

$sin(\pi+\alpha)=-sin(\alpha)$
$cos(\pi+\alpha)=-cos(\alpha)$
$tan(\pi+\alpha)=tan(\alpha)$

 2.

$sin(-\alpha)=-sin(\alpha)$
$cos(-\alpha)=cos(\alpha)$
$tan(-\alpha)=-tan(\alpha)$

 3.

$sin(\pi-\alpha)=sin(\alpha)$
$cos(\pi-\alpha)=-cos(\alpha)$
$tan(\pi-\alpha)=-tan(\alpha)$

4.

$sin(\frac{\pi}{2}-\alpha)=cos(\alpha)$
$cos(\frac{\pi}{2}-\alpha)=sin(\alpha)$

5.

$sin(\frac{\pi}{2}+\alpha)=cos(\alpha)$
$cos(\frac{\pi}{2}+\alpha)=-sin(\alpha)$

和差公式:

1.

$cos(\alpha+\beta)=cos(\alpha) \times cos(\beta)-sin(\alpha) \times sin(\beta)$
$cos(\alpha-\beta)=cos(\alpha) \times cos(\beta)+sin(\alpha) \times sin(\beta)$

2.

$sin(\alpha+\beta)=sin(\alpha) \times cos(\beta)+cos(\alpha) \times sin(\beta)$
$sin(\alpha-\beta)=sin(\alpha) \times cos(\beta)-cos(\alpha) \times sin(\beta)$

3.

$tan(\alpha+\beta)=\frac{tan(\alpha)+tan(\beta)}{1-tan(\alpha) \times tan(\beta)}$
$tan(\alpha-\beta)=\frac{tan(\alpha)-tan(\beta)}{1+tan(\alpha) \times tan(\beta)}$

4.

$sin(2 \times \alpha)=2sin(\alpha)\times cos(\alpha)$
$cos(2 \times \alpha)=cos(\alpha)^2-sin(\alpha)^2=2 \times cos(\alpha)^2-1=1-2 \times sin(\alpha)^2$
$tan(2 \times \alpha)=\frac{2 \times tan(\alpha)}{1-tan(\alpha)^2}$

5.

$sin(\frac{\alpha}{2})=±\sqrt{\frac{1-cos\alpha}{2}}$

$cos(\frac{\alpha}{2})=±\sqrt{\frac{1+cos\alpha}{2}}$

$tan(\frac{\alpha}{2})=\frac{sin(\alpha)}{1+cos(\alpha)}=\frac{1-cos(\alpha)}{sin(\alpha)}=±\sqrt{\frac{1-cos(\alpha)}{1+cos(\alpha)}}$

竞赛

$sin(\alpha)+sin(2\times\alpha)+sin(3\times\alpha)+...+sin(n\times\alpha)=\frac{sin(\frac{n}{2}\alpha)\times sin(\frac{n+1}{2}\alpha)}{sin(\frac{\alpha}{2})}$
$cos(\alpha)+cos(2\times\alpha)+cos(3\times\alpha)+...+cos(n\times\alpha)=\frac{sin(\frac{n+1}{2}\alpha)+sin(n\times\alpha)-sin(\alpha)}{2\times sin(\alpha)}$