博主退役了。
博主去学文化课了。
博主发现文化课好难。
博主学不动了。
诱导公式
先给出一张重要的图
（快感谢我这次用 Geogebra 画图而不是 MS-Paint）
第一组
[sin (alpha+kcdot 2pi)=sinalpha(kin Z)
]
[cos (alpha+kcdot 2pi)=cosalpha(kin Z)
]
[ an (alpha+kcdot 2pi)= analpha(kin Z)
]
第二组
[sin(alpha+pi)=-sin alpha
]
[cos(alpha+pi)=-cosalpha
]
[ an(alpha+pi)= analpha
]
第三组
[sin(-alpha)=-sinalpha
]
[cos(-alpha)=cosalpha
]
[ an(-alpha)=- analpha
]
第四组
[sin(pi-alpha)=sin alpha
]
[cos(pi-alpha)=cosalpha
]
[ an(pi-alpha)=- analpha
]
以上四组根据上图显然。
第五组
[sin(frac{pi}{2}-alpha)=cosalpha
]
[cos(frac{pi}{2}-alpha)=sinalpha
]
[ an(frac{pi}{2}-alpha)=frac{1}{ analpha}
]
这是常识（雾）。
第六组
[sin(alpha+frac{pi}{2})=cosalpha
]
[cos(alpha+frac{pi}{2})=-sinalpha
]
[ an(alpha+frac{pi}{2})=-frac{1}{ analpha}
]
证明：
[egin{aligned}
sin(alpha+frac{pi}{2})
&=cos(frac{pi}{2}-alpha-frac{pi}{2})\
&=cos(-alpha)\
&=cos alpha
end{aligned}]
[egin{aligned}
cos(alpha+frac{pi}{2})
&=sin(frac{pi}{2}-alpha-frac{pi}{2})\
&=sin(-alpha)\
&=-sin alpha
end{aligned}]
[egin{aligned}
an(alpha+frac{pi}{2})
&=frac{sin(alpha+frac{pi}{2})}{cos(alpha+frac{pi}{2})}\
&=frac{cosalpha}{-sinalpha}\
&=-frac{1}{ analpha}
end{aligned}]
和差角公式
[cos(alpha-eta)=cosalphacoseta+sinalphasineta
]
[cos(alpha+eta)=cosalphacoseta-sinalphasineta
]
[sin(alpha-eta)=sinalphacoseta-cosalphasineta
]
[sin(alpha+eta)=sinalphacoseta+cosalphasineta
]
证明：如图所示，(A(cosalpha,sinalpha))，(B(coseta,sineta)) 。
此处需要用到向量点积。
[egin{aligned}
cos(alpha-eta)&=
frac{vec{OA}cdotvec{OB}}{|vec{OA}||vec{OB}|}\
&=frac{x_Ax_B+y_Ay_B}{1 imes1}\
&=cosalphacoseta+sinalphasineta
end{aligned}]
[egin{aligned}
cos(alpha+eta)
&=cosleft(alpha-(-eta)
ight)\
&=cosalphacos(-eta)+sinalphasin(-eta)\
&=cosalphacoseta-sinalphasineta\
end{aligned}]
[egin{aligned}
sin(alpha-eta)
&=cos(frac{pi}{2}-(alpha-eta))\
&=cos((frac{pi}{2}-alpha)+eta))\
&=cos(frac{pi}{2}-alpha)coseta-sin(frac{pi}{2}-alpha)sineta\
&=sinalphacoseta-cosalphasineta\
end{aligned}]
[egin{aligned}
sin(alpha+eta)
&=cos(frac{pi}{2}-(alpha+eta))\
&=cos((frac{pi}{2}-alpha)-eta))\
&=cos(frac{pi}{2}-alpha)coseta+sin(frac{pi}{2}-alpha)sineta\
&=sinalphacoseta+cosalphasineta\
end{aligned}]
二倍角公式
[cos(2alpha)=cos^2alpha-sin^2alpha=2cos^2alpha-1=1-2sin^2alpha
]
[sin(2alpha)=2cosalphasinalpha
]
将 (2alpha) 带入加法公式即可。
[ an(2alpha)=frac{2 analpha}{1- an^2alpha}
]
证明：
[egin{aligned}
an(2alpha)
&=frac{sin{2alpha}}{cos{2alpha}}\
&=frac{2sinalphacosalpha}{cos^2alpha-sin^2alpha}\
&=frac{frac{2sinalpha}{cosalpha}}{1-frac{sin^2alpha}{cos^2alpha}}\
&=frac{2 analpha}{1- an^2alpha}
end{aligned}
]
半角公式
[cos(frac{alpha}{2})=pmsqrt{frac{1+cosalpha}{2}}
]
[sin(frac{alpha}{2})=pmsqrt{frac{1-cosalpha}{2}}
]
证明：
[egin{aligned}
cosalpha&=2cos^2(frac{alpha}{2})-1\
cos(frac{alpha}{2})&=pmsqrt{frac{1+cosalpha}{2}}\
end{aligned}]
[egin{aligned}
cosalpha&=1-2sin^2(frac{alpha}{2})\
sin(frac{alpha}{2})&=pmsqrt{frac{1-cosalpha}{2}}\
end{aligned}]
[egin{aligned}
an(frac{alpha}{2})&=frac{sin(frac{alpha}{2})}{cos(frac{alpha}{2})}\
&=frac{pmsqrt{frac{1-cosalpha}{2}}}{pmsqrt{frac{1+cosalpha}{2}}}\
&=pmsqrt{frac{1-cosalpha}{1+cosalpha}}
end{aligned}]
正弦定理
[frac{a}{sin A}=frac{b}{sin B}=frac{c}{sin C}=2R
]
其中 (R) 是外接圆半径。
余弦定理
[a^2=b^2+c^2-2bccos A
]