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[begin{align}
l=alpha cdot r qquad S=frac12alpha r^2=frac12 l , r
end{align}
]
[(sinalpha + cosalpha)^2 = 1 + 2sinalpha cosalpha qquad
(sinalpha - cosalpha)^2 = 1 - 2sinalpha cosalpha
]
[begin{align}
sin (2k pi + alpha) &= sin alpha &
cos (2k pi + alpha) &= cos alpha &
tan (2k pi + alpha) &= tan alpha\
end{align}
]
[begin{align}
sin ( pi - alpha) &= color{red}sin alpha &
cos ( pi - alpha) &= -cos alpha &
tan ( pi - alpha) &= -tan alpha\
sin ( - alpha) &= -sin alpha &
cos ( - alpha) &= color{red}cos alpha &
tan ( - alpha) &= -tan alpha\
sin ( pi + alpha) &= -sin alpha &
cos ( pi + alpha) &= -cos alpha &
tan ( pi + alpha) &= color{red}tan alpha\
end{align}
]
[begin{align}
sin ( fracpi2 - alpha) &= cos alpha &
sin ( fracpi2 + alpha) &= cos alpha &\
cos ( fracpi2 - alpha) &= sin alpha &
cos ( fracpi2 + alpha) &= -sin alpha &\
tan ( fracpi2 - alpha) &= cot alpha &
tan ( fracpi2 + alpha) &= -cot alpha &\
cot ( fracpi2 - alpha) &= tan alpha &
cot ( fracpi2 + alpha) &= -tan alpha &\
end{align}
]
[begin{align}
sin ( frac{3pi}2 - alpha) &= -cos alpha &
sin ( frac{3pi}2 + alpha) &= -cos alpha &\
cos ( frac{3pi}2 - alpha) &= -sin alpha &
cos ( frac{3pi}2 + alpha) &= sin alpha &\
tan ( frac{3pi}2 - alpha) &= -cot alpha &
tan ( frac{3pi}2 + alpha) &= -cot alpha &\
cot ( frac{3pi}2 - alpha) &= tan alpha &
cot ( frac{3pi}2 + alpha) &= -tan alpha &\
end{align}
]
[begin{align}
sin (alpha + beta) &= sinalpha cosbeta + cosalpha sinbeta &
sin (alpha - beta) &= sinalpha cosbeta - cosalpha sinbeta \
cos (alpha + beta) &= cosalpha cosbeta - sinalpha sinbeta &
cos (alpha - beta) &= cosalpha cosbeta + sinalpha sinbeta \
tan (alpha + beta) &= frac{tanalpha + tanbeta}{1 - tanalpha tanbeta} &
tan (alpha - beta) &= frac{tanalpha - tanbeta}{1 + tanalpha tanbeta} \
end{align}
]
[begin{align}
sin 2alpha &= 2sinalpha cosalpha \
cos 2alpha &= cos^2alpha - sin^2alpha = 2cos^2alpha-1 = 1-2sin^2alpha \
tan 2alpha &= frac{2tanalpha}{1 - tan^2alpha} \
end{align}
]
[begin{align}
sin fracalpha2 &= pm sqrt{frac{1 - cosalpha}{2}} \
cos fracalpha2 &= pm sqrt{frac{1 + cosalpha}{2}} \
tan fracalpha2 &= pm sqrt{frac{1 - cosalpha}{1 + cosalpha}} \
tan fracalpha2 &= frac{sinalpha}{1 + cosalpha}= frac{1 - cosalpha}{sinalpha} \
end{align}
]
[begin{align}
sinalpha cosbeta&=frac12left[sin(alpha+beta)+sin(alpha-beta)right] &
cosalpha sinbeta&=frac12left[sin(alpha+beta)-sin(alpha-beta)right] \
cosalpha cosbeta&=frac12left[cos(alpha+beta)+cos(alpha-beta)right] &
sinalpha sinbeta&=-frac12left[cos(alpha+beta)-cos(alpha-beta)right] \
end{align}
]
[begin{align}
sinalpha + sinbeta&=2sinfrac{alpha+beta}2 sinfrac{alpha-beta}2 &
sinalpha - sinbeta&=2sinfrac{alpha+beta}2 sinfrac{alpha-beta}2 \
cosalpha + cosbeta&=2cosfrac{alpha+beta}2 cosfrac{alpha-beta}2 &
cosalpha - cosbeta&=-2cosfrac{alpha+beta}2 cosfrac{alpha-beta}2 \
end{align}
]
[asinalpha + bcosalpha=sqrt{a^2+b^2}sin(alpha+varphi)\
left(cosvarphi=frac a{sqrt{a^2+b^2}}, sinvarphi=frac b{sqrt{a^2+b^2}}, tanvarphi=frac baright)
]
[begin{align}
sin 3alpha &= 3sinalpha - 4sin^3alpha = 4sinalpha sin(fracpi3 - alpha) sin(fracpi3 + alpha) \
cos 3alpha &= 4cos^3alpha - 3cosalpha = 4cosalpha cos(fracpi3 - alpha) cos(fracpi3 + alpha) \
cos 3alpha &= frac{3tanalpha-tan^3alpha}{1-3tan^2alpha} = 4tanalpha tan(fracpi3 - alpha) tan(fracpi3 + alpha) \
end{align}
]
[begin{align}
tanalpha + cotalpha &= frac1{sinalpha cosalpha} = frac2{sin 2alpha}\
tanalpha - cotalpha &= frac{sin^2alpha - cos^2alpha}{sinalpha cosalpha} = frac{-2 cos{2alpha}}{sin 2alpha} = -2 cot{2alpha}\
end{align}
]
[begin{align}
sinalpha &= frac{2tandisplaystylefrac{alpha}{2}}{1 + tan^2displaystylefrac{alpha}{2}} &
cosalpha &= frac{1 - tan^2displaystylefrac{alpha}{2}}{1 + tan^2displaystylefrac{alpha}{2}} &
tanalpha &= frac{2tandisplaystylefrac{alpha}{2}}{1 - tan^2displaystylefrac{alpha}{2}} \
end{align}
]
[begin{align}
sin^2alpha &= frac{1 - cos2alpha}2 &
cos^2alpha &= frac{1 + cos2alpha}2 &
sinalpha cosalpha &= frac{sin 2alpha}2 \
end{align}
]
在三角形(ABC)中，(A + B + C = 2pi)，(A、B、Cnot=displaystylefracpi2)
[sinfrac A2 = cosfrac{B + C}2 qquad tanfrac A2 = cotfrac{B + C}2\
tan A + tan B + tan C = tan A tan B tan C
]
切比雪夫多项式
[g_{n+1}(x)=2xg_n(x)-g_{n-1}(x)
]
[begin{align}
cos 2alpha &= 2cos^2alpha - 1\
cos 3alpha &= 4cos^3alpha - 3cosalpha\
cos 4alpha &= 8cos^4alpha - 8cos^2alpha + 1\
cos 5alpha &= 16cos^5alpha - 20cos^3alpha + 5cosalpha\
end{align}
]
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