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  • 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

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

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  • 原文地址:https://www.cnblogs.com/zhouykblog/p/10255600.html
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