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.

$cos(alpha+eta)=cos(alpha) imes cos(eta)-sin(alpha) imes sin(eta)$
$cos(alpha-eta)=cos(alpha) imes cos(eta)+sin(alpha) imes sin(eta)$

2.

$sin(alpha+eta)=sin(alpha) imes cos(eta)+cos(alpha) imes sin(eta)$
$sin(alpha-eta)=sin(alpha) imes cos(eta)-cos(alpha) imes sin(eta)$

3.

$tan(alpha+eta)=frac{tan(alpha)+tan(eta)}{1-tan(alpha) imes tan(eta)}$
$tan(alpha-eta)=frac{tan(alpha)-tan(eta)}{1+tan(alpha) imes tan(eta)}$

4.

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

5.

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

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

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

竞赛

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