诱导公式:奇变偶不变,符号看象限
无敌六边形:
其中有三组关系:
- 边上的三角函数两边相乘等于中间
- 染了色的三角形上面两个三角函数相乘等于下面的
- 相对的三角函数是倒数关系
和差角公式:
- (sin(alphapmeta)=sinalphacosetapmsinetacosalpha)
- (cos(alphapmeta)=cosalphacosetampsinalphasineta)
- ( an(alphapmeta)=dfrac{ analphapm aneta}{1mp analpha aneta})
二倍角公式:
- (sin2alpha=2sinalphacosalpha)
- (egin{aligned}cos2alpha&=cos^2alpha-sin^2alpha\&=2cos^2alpha-1\&=1-2sin^2alphaend{aligned})
- ( an2alpha=dfrac{2 analpha}{1- an^2alpha})
三倍角公式:
- (sin3alpha=3sinalphacos^2alpha-sin^3alpha)
- (cos3alpha=cos^3alpha-3sin^2alphacosalpha)
半角公式:
- (sindfrac{alpha}2=pmsqrt{dfrac{1-cosalpha}2}),符号看象限
- (cosdfrac{alpha}2=pmsqrt{dfrac{1+cosalpha}2}),符号看象限
- ( andfrac{alpha}2=pmsqrt{dfrac{1-cosalpha}{1+cosalpha}}),符号看象限
- (egin{aligned} andfrac{alpha}2&=dfrac{sinalpha}{1+cosalpha}\&=dfrac{1-cosalpha}{sinalpha}\end{aligned})
点鞭炮公式:
[cos hetacos2 hetacos4 hetacdotscos2^n heta=sum_{i=0}^ncos2^i heta=dfrac{sin2^{n+1}alpha}{2^{n+1}sinalpha}
]
降幂公式:
- (sinalphacosalpha=dfrac{sin2alpha}2)
- (sin^2alpha=dfrac{1-cos2alpha}2)
- (cos^2alpha=dfrac{1+cos2alpha}2)
(lambda) 等分点:
若 (overrightarrow{p_1}overrightarrow p=lambdaoverrightarrow poverrightarrow{p_2}),其中 (p_1(a_1,a_2,cdots,a_n),p_2(b_1,b_2,cdots,b_n))((n) 维坐标,特殊情况是 (n=2) 或 (n=3))
则
[p=left(dfrac{a_1+lambda b_1}{1+lambda},dfrac{a_2+lambda b_2}{1+lambda},cdots,dfrac{a_n+lambda b_n}{1+lambda}
ight)
]
辅助角公式:
[asin heta+bcos heta=sqrt{a^2+b^2}sin( heta+varphi)
]
其中 ( anvarphi=dfrac ba)
和差化积 & 积化和差 公式:
- (sinalphacoseta=dfrac 12(sin(alpha+eta)+sin(alpha-eta)))
- (cosalphasineta=dfrac 12(sin(alpha+eta)-sin(alpha-eta)))
- (cosalphacoseta=dfrac 12(cos(alpha+eta)+cos(alpha-eta)))
- (sinalphasineta=-dfrac 12(cos(alpha+eta)-cos(alpha-eta)))
- (sin heta+sinvarphi=2sindfrac{ heta+varphi}2cosdfrac{ heta-varphi}2)
- (sin heta-sinvarphi=2cosdfrac{ heta+varphi}2sindfrac{ heta-varphi}2)
- (cos heta+cosvarphi=2cosdfrac{ heta+varphi}2cosdfrac{ heta-varphi}2)
- (cos heta-cosvarphi=-2sindfrac{ heta+varphi}2sindfrac{ heta-varphi}2)
- (sin heta+cosvarphi) 这种可以用辅助角公式。