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Cоотношения
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sin2(α)+cos2(α)=1
| tg(α)ctg(α)=1
| sin(-α)=-sin(α) |
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cos(-α)=cos(α)
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sin(α+2πn)=sin(α)
| cos(α+2πn)=cos(α)
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tg(α)=sin(α)/cos(α)
| ctg(α)=cos(α)/sin(α)
| tg(-α)=-tg(α) | |
ctg(-α)=-ctg(α)
| tg(α+πn)=tg(α)
| ctg(α+πn)=ctg(α) | |
1+tg2(α)=1/cos2(α)
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1+ctg2(α)=1/sin2(α)
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sec(α)=1/cos(α)
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cosec(α)=1/sin(α)
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Cвязь между тригонометрическими функциями одного аргумента
sin(α)=±(1-cos2(α))1/2=±tg(α)/(1+tg2(α))1/2=±1/(1+ctg2(α))1/2
cos(α)=±(1-sin2(α))1/2=±ctgα/(1+ctg2(α))1/2=(α)1/(1+tg2(α))1/2
tg(α)=±1/ctg(α)=±sin(α)/(1-sin2(α))1/2=±(1+cos2(α))1/2/cos(α)
ctg(α)=±1/tg(α)=±cos(α)/(1-cos2(α))1/2=±(1+sin2(α))1/2/sin(α)
Формулы для суммы и разности аргументов
sin(α±β)=sin(α)cos(β)±sin(β)cos(α)
cos(α±β)=cos(α)cos(β)±sin(β)sin(α)
tg(α±β)=(tg(α)±tg(β))/(1±tg(-β)tg(α))
ctg(α±β)=-(ctg(-α)ctg(β)±1)/(ctg(β)±ctg(α))
Формулы двойных, тройных и половинных аргументов
sin(2α)=2sin(α)cos(α)=2tg(α)/(1+tg2α)
cos(2α)=cos2(α)-sin2(α)=1-2sin2α=(1-tg2α)/(1+tg2α)
tg(2α)=2tg(α)/(1-tg2(α))
ctg(2α)=(ctg2(α)-1)/(2ctg(α))
sin(3α)=3sin(α)-4sin3(α)
cos(3α)=4cos3(α)-3cos(α)
tg(3α)=(3tg(α)-tg3(α))/(1-3tg2(α))
ctg(3α)=(ctg3(α)-3ctg(α))/(3ctg2(α)-1)
sin2(α/2)=(1-cos(α))/2
cos2(α/2)=(1+cos(α))/2
tg(α/2)=sin(α)/(1+cos(α))=(1-cos(α))/sin(α)=±((1-cos(α))/(1+cos(α)))1/2
ctg(α/2)=sin(α)/(1-cos(α))=(1+cos(α))/sin(α)=±((1+cos(α))/(1-cos(α)))1/2
Формулы преобразования произведения в сумму и обратно
sin(α)cos(β)=1/2(sin(α+β)+sin(α-β))
cos(α)cos(β)=1/2(cos(α+β)+cos(α-β))
sin(α)sin(β)=1/2(cos(α-β)-cos(α-β))
sin(α)+sin(β)=2sin((α+β)/2)cos((α-β)/2)
sin(α)-sin(β)=2sin((α-β)/2)cos((α+β)/2)
cos(α)+cos(β)=2cos((α+β)/2)cos((α-β)/2)
cos(α)-cos(β)=-2sin((α+β)/2)sin((α-β)/2)
tg(α)±tg(β)=(sin(α±β))/(cos(β)cos(α))
ctg(α)±ctg(β)=(sin(α±β))/(sin(β)sin(α))
tg(α)+ctg(β)=(cos(α-β))/(sin(β)cos(α))
tg(α)-ctg(β)=-(cos(α+β))/(sin(β)cos(α))
tg(α)+ctg(α)=-2cosec(2α)
tg(α)-ctg(α)=-2ctg(2α)
tg2(α)-tg2(β)=sin(α+β)sin(α-β)/(cos2(α)cos2(β))
1+tg(α)tg(β)=(cos(α-β))/(cos(β)cos(α))
1-tg(α)tg(β)=(cos(α+β))/(cos(β)cos(α))
ctg(α)ctg(β)+1=(cos(α-β))/(sin(β)sin(α))
ctg(α)ctg(β)-1=(cos(α+β))/(sin(β)sin(α))
tg2(α)-tg2(β)=sin(α+β)sin(α-β)/(sin2(α)sin2(β))
tg2(α)-sin2(α)=tg2(α)sin2(α)
ctg2(α)-cos2(α)=ctg2(α)cos2(α)
sin3(α)=(3sin(α)-sin(3α))/4
cos3(α)=(3cos(α)+cos(3α))/4
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