Формулы тригонометрии

Основные тождества

tgα=sin⁡αcosα=1ctgα\displaystyle{ tg\alpha = \frac{\sin\alpha}{cos\alpha} = \frac{1}{ctg\alpha} } ctgα=cos⁡αsinα=1tgα\displaystyle{ ctg\alpha = \frac{\cos\alpha}{sin\alpha} = \frac{1}{tg\alpha} } sin2α+cos2α=1\displaystyle{ sin^2\alpha+cos^2\alpha=1 } 1+tg2α=1cos2α\displaystyle{ 1 + tg^2\alpha = \frac{1}{cos^2\alpha} } 1+ctg2α=1sin2α\displaystyle{ 1 + ctg^2\alpha = \frac{1}{sin^2\alpha} } tgα⋅ctgα=1\displaystyle{ tg\alpha \cdot ctg\alpha = 1 }

Формулы двойного угла

sin(2α)=2⋅cosα⋅sinα\displaystyle{ sin(2\alpha) = 2\cdot cos\alpha \cdot sin\alpha } sin2α=2⋅tgα1+tg2α=2⋅ctgα1+ctg2α=2tgα+ctgα\displaystyle{ sin2\alpha = \frac{2\cdot tg\alpha}{1+tg^2\alpha} = \frac{2\cdot ctg\alpha}{1+ctg^2\alpha} = \frac{2}{tg\alpha+ctg\alpha} } cos(2α)=cos2α−sin2α=2⋅cos2α−1=1−2⋅sin2α\displaystyle{ cos(2\alpha) = cos^2\alpha-sin^2\alpha = 2\cdot cos^2\alpha-1 = 1-2\cdot sin^2\alpha } cos(2α)=1−tg2α1+tg2α=ctg2α−1ctg2α+1=ctgα−tgαctgα+tgα\displaystyle{ cos(2\alpha) = \frac{1-tg^2\alpha}{1+tg^2\alpha} = \frac{ctg^2\alpha-1}{ctg^2\alpha+1}= \frac{ctg\alpha-tg\alpha}{ctg\alpha+tg\alpha} } tg(2α)=2⋅tgα1−tg2α=2⋅ctgαctg2α−1=2ctgα−tgα\displaystyle{ tg(2\alpha) = \frac{2\cdot tg\alpha}{1-tg^2\alpha} = \frac{2\cdot ctg\alpha}{ctg^2\alpha-1}= \frac{2}{ctg\alpha-tg\alpha} } ctg(2α)=ctg2α−12⋅ctgα=ctgα−tgα2\displaystyle{ ctg(2\alpha) = \frac{ctg^2\alpha-1}{2\cdot ctg\alpha} = \frac{ctg\alpha-tg\alpha}{2} }

Формулы тройного угла

sin(3α)=3⋅sinα−4⋅sin3α\displaystyle{ sin(3\alpha)=3\cdot sin\alpha-4\cdot sin^3\alpha } cos(3α)=4⋅cos3α−3⋅cosα\displaystyle{ cos(3\alpha)=4\cdot cos^3\alpha-3\cdot cos\alpha } tg(3α)=3⋅tgα−tg3α1−3⋅tg2α\displaystyle{ tg(3\alpha)= \frac{3\cdot tg\alpha-tg^3\alpha}{1-3\cdot tg^2\alpha} } ctg(3α)=ctg3α−3⋅ctgα3⋅ctg2α−1\displaystyle{ ctg(3\alpha) = \frac{ctg^3\alpha-3\cdot ctg\alpha}{3\cdot ctg^2\alpha-1} }

Формулы понижения степени

sin2α=1−cos(2α)2\displaystyle{ sin^2\alpha = \frac{1 - cos(2\alpha)}{2} } cos2α=1+cos(2α)2\displaystyle{ cos^2\alpha = \frac{1 + cos(2\alpha)}{2} } tg2α=1−cos(2α)1+cos(2α)\displaystyle{ tg^2\alpha= \frac{1-cos(2\alpha)}{1+cos(2\alpha)} } ctg2α=1+cos(2α)1−cos(2α)\displaystyle{ ctg^2\alpha= \frac{1+cos(2\alpha)}{1-cos(2\alpha)} } (sinα−cosα)2=1−sin(2α)\displaystyle{ (sin\alpha - cos\alpha)^2= 1-sin(2\alpha) } (sinα+cosα)2=1+sin(2α)\displaystyle{ (sin\alpha + cos\alpha)^2= 1+sin(2\alpha) }

Формулы понижения степени

sin3(α)=3⋅sin(α)−sin(3α)4\displaystyle{ sin^3(\alpha)= \frac{3\cdot sin(\alpha)-sin(3\alpha)}{4} } cos3(α)=3⋅cos(α)+cos(3α)4\displaystyle{ cos^3(\alpha)= \frac{3\cdot cos(\alpha)+cos(3\alpha)}{4} } tg3(α)=3⋅sin(α)−sin(3α)3⋅cos(α)+cos(3α)\displaystyle{ tg^3(\alpha)= \frac{3\cdot sin(\alpha)-sin(3\alpha)}{3\cdot cos(\alpha)+cos(3\alpha)} } ctg3(α)=3⋅cos(α)+cos(3α)3⋅sin(α)−sin(3α)\displaystyle{ ctg^3(\alpha)= \frac{3\cdot cos(\alpha)+cos(3\alpha)}{3\cdot sin(\alpha)-sin(3\alpha)} }

Формулы понижения степени

sin4(α)=3−4⋅cos(2α)+cos(4α)8\displaystyle{ sin^4(\alpha)= \frac{3-4\cdot cos(2\alpha)+cos(4\alpha)}{8} } cos4(α)=3+4⋅cos(2α)+cos(4α)8\displaystyle{ cos^4(\alpha)= \frac{3+4\cdot cos(2\alpha)+cos(4\alpha)}{8} } sin5(α)=10⋅sin(α)−5⋅sin(3α)+sin(5α)16\displaystyle{ sin^5(\alpha)= \frac{10\cdot sin(\alpha) - 5\cdot sin(3\alpha) + sin(5\alpha) }{16} } cos5(α)=10⋅cos(α)+5⋅cos(3α)+cos(5α)16\displaystyle{ cos^5(\alpha)= \frac{10\cdot cos(\alpha) + 5\cdot cos(3\alpha) + cos(5\alpha) }{16} }

Формулы половинного аргумента

sin(α2)=±1−cosα2\displaystyle{ sin \left(\frac{\alpha}{2}\right) = \pm \sqrt{ \frac{1-cos\alpha}{2} } } cos(α2)=±1+cosα2\displaystyle{ cos \left(\frac{\alpha}{2}\right) = \pm \sqrt{ \frac{1+cos\alpha}{2} } } tg(α2)=1−cosαsinα=sinα1+cosα\displaystyle{ tg \left(\frac{\alpha}{2}\right)= \frac{1-cos\alpha}{sin\alpha}= \frac{sin\alpha}{1+cos\alpha} } ctg(α2)=1+cosαsinα=sinα1−cosα\displaystyle{ ctg \left(\frac{\alpha}{2}\right) = \frac{1+cos\alpha}{sin\alpha}= \frac{sin\alpha}{1-cos\alpha} }

Формулы понижения степени половинного аргумента

sin2(α2)=1−cosα2\displaystyle{ sin^2 \left( \frac{\alpha}{2} \right)= \frac{1-cos\alpha}{2} } cos2(α2)=1+cosα2\displaystyle{ cos^2 \left( \frac{\alpha}{2} \right)= \frac{1+cos\alpha}{2} } tg2(α2)=1−cosα1+cosα\displaystyle{ tg^2 \left( \frac{\alpha}{2} \right)= \frac{1-cos\alpha}{1+cos\alpha} } ctg2(α2)=1+cosα1−cosα\displaystyle{ ctg^2 \left( \frac{\alpha}{2} \right)= \frac{1+cos\alpha}{1-cos\alpha} }

Формулы сложения

sin(α+β)=sinα⋅cosβ+cosα⋅sinβ\displaystyle{ sin(\alpha + \beta) = sin \alpha \cdot cos \beta + cos \alpha \cdot sin \beta } cos(α+β)=cosα⋅cosβ−sinα⋅sinβ\displaystyle{ cos(\alpha + \beta) = cos \alpha \cdot cos \beta - sin \alpha \cdot sin \beta } tg(α+β)=tgα+tgβ1−tgα⋅tgβ\displaystyle{ tg(\alpha + \beta)= \frac{tg \alpha + tg \beta}{1 - tg \alpha \cdot tg \beta} } ctg(α+β)=ctgα⋅ctgβ−1ctgα+ctgβ\displaystyle{ ctg(\alpha + \beta)= \frac{ctg \alpha \cdot ctg \beta -1}{ctg \alpha + ctg \beta} }

Формулы вычитания

sin(α−β)=sinα⋅cosβ−cosα⋅sinβ\displaystyle{ sin(\alpha - \beta) = sin \alpha \cdot cos \beta - cos \alpha \cdot sin \beta } cos(α−β)=cosα⋅cosβ+sinα⋅sinβ\displaystyle{ cos(\alpha - \beta) = cos \alpha \cdot cos \beta + sin \alpha \cdot sin \beta } tg(α−β)=tgα−tgβ1+tgα⋅tgβ\displaystyle{ tg(\alpha - \beta)= \frac{tg \alpha - tg \beta}{1 + tg \alpha \cdot tg \beta} } ctg(α−β)=ctgα⋅ctgβ+1ctgα−ctgβ\displaystyle{ ctg(\alpha - \beta)= \frac{ctg \alpha \cdot ctg \beta +1}{ctg \alpha - ctg \beta} }

Формулы преобразования суммы в формулы произведения

sinα+sinβ=2⋅sin(α+β2)⋅cos(α−β2)\displaystyle{ sin\alpha + sin\beta = 2\cdot sin \left( \frac{\alpha + \beta }{2} \right) \cdot cos \left( \frac{\alpha - \beta }{2} \right) } cosα+cosβ=2⋅cos(α+β2)⋅cos(α−β2)\displaystyle{ cos\alpha + cos\beta = 2\cdot cos \left( \frac{\alpha + \beta }{2} \right) \cdot cos \left( \frac{\alpha - \beta }{2} \right) } tgα+tgβ=sin(α+β)cosα⋅cosβ\displaystyle{ tg\alpha + tg\beta = \frac{sin(\alpha + \beta) }{cos \alpha \cdot cos \beta} } ctgα+ctgβ=sin(α+β)cosα⋅cosβ\displaystyle{ ctg\alpha + ctg\beta = \frac{sin(\alpha + \beta) }{cos \alpha \cdot cos \beta} }

Формулы преобразования разностив формулы произведения

sinα−sinβ=2⋅sin(α−β2)⋅cos(α+β2)\displaystyle{ sin\alpha - sin\beta = 2\cdot sin \left( \frac{\alpha - \beta }{2} \right) \cdot cos \left( \frac{\alpha + \beta }{2} \right) } cosα−cosβ=−2⋅sin(α+β2)⋅sin(α−β2)\displaystyle{ cos\alpha - cos\beta = -2\cdot sin \left( \frac{\alpha + \beta }{2} \right) \cdot sin \left( \frac{\alpha - \beta }{2} \right) } tgα−tgβ=sin(α−β)cosα⋅cosβ\displaystyle{ tg\alpha - tg\beta = \frac{sin(\alpha - \beta) }{cos \alpha \cdot cos \beta} } ctgα−ctgβ=−sin(α+β)sinα⋅sinβ\displaystyle{ ctg\alpha - ctg\beta = -\frac{sin(\alpha + \beta) }{sin \alpha \cdot sin \beta} }

Формулы преобразования суммы

sin(α)+cos(α)=2⋅sin(α+π4)\displaystyle{ sin(\alpha) + cos(\alpha) = \sqrt{2}\cdot sin\left( \alpha+\frac{\pi}{4} \right) } sin(α)−cos(α)=2⋅sin(α−π4)\displaystyle{ sin(\alpha) - cos(\alpha) = \sqrt{2}\cdot sin\left( \alpha-\frac{\pi}{4} \right) } arcsin(x)+arccos(x)=π2\displaystyle{ arcsin(x) + arccos(x) = \frac{\pi}{2} } arctg(x)+arcctg(x)=π2\displaystyle{ arctg(x) + arcctg(x) = \frac{\pi}{2} } Asin(α)+Bcos(α)=A2+B2(sin(α+arccos(AA2+B2)))\displaystyle{ Asin(\alpha) + Bcos(\alpha) = \sqrt{A^2+B^2}(sin(\alpha + arccos\left( \frac{A}{\sqrt{A^2+B^2}} \right))) } Asin(α)−Bcos(α)=A2+B2(sin(α−arccos(AA2+B2)))\displaystyle{ Asin(\alpha) - Bcos(\alpha) = \sqrt{A^2+B^2}(sin(\alpha - arccos\left( \frac{A}{\sqrt{A^2+B^2}} \right))) }

Формулы преобразования произведения в формулы суммы и разности

sinα⋅sinβ=cos(α−β)−cos(α+β)2\displaystyle{ sin\alpha \cdot sin\beta = \frac{cos(\alpha - \beta)-cos(\alpha + \beta)}{2} } sinα⋅cosβ=sin(α−β)+sin(α+β)2\displaystyle{ sin\alpha \cdot cos\beta = \frac{sin(\alpha - \beta)+sin(\alpha + \beta)}{2} } cosα⋅cosβ=cos(α−β)+cos(α+β)2\displaystyle{ cos\alpha \cdot cos\beta = \frac{cos(\alpha - \beta)+cos(\alpha + \beta)}{2} } tgα⋅tgβ=cos(α−β)−cos(α+β)cos(α−β)+cos(α+β)=tgα+tgβctgα+ctgβ\displaystyle{ tg\alpha \cdot tg\beta = \frac{cos(\alpha - \beta)-cos(\alpha + \beta)}{cos(\alpha - \beta)+cos(\alpha + \beta)} = \frac{tg\alpha + tg\beta}{ctg\alpha + ctg\beta} } ctgα⋅ctgβ=cos(α−β)+cos(α+β)cos(α−β)−cos(α+β)=ctgα+ctgβtgα+tgβ\displaystyle{ ctg\alpha \cdot ctg\beta = \frac{cos(\alpha - \beta)+cos(\alpha + \beta)}{cos(\alpha - \beta)-cos(\alpha + \beta)} = \frac{ctg\alpha + ctg\beta}{tg\alpha + tg\beta} } tgα⋅ctgβ=sin(α−β)+sin(α+β)sin(α+β)−sin(α−β)\displaystyle{ tg\alpha \cdot ctg\beta = \frac{sin(\alpha - \beta)+sin(\alpha + \beta)}{sin(\alpha + \beta)-sin(\alpha - \beta)} }

Формулы преобразования произведения функций в степени

sin2(α)⋅cos2(α)=1−cos(4α)8\displaystyle{ sin^2(\alpha)\cdot cos^2(\alpha) = \frac{1 - cos(4\alpha) }{8} } sin3(α)⋅cos3(α)=3⋅sin(2α)−sin(6α)32\displaystyle{ sin^3(\alpha)\cdot cos^3(\alpha) = \frac{ 3\cdot sin(2\alpha) - sin(6\alpha) }{32} } sin4(α)⋅cos4(α)=3−4⋅cos(4α)+cos(8α)128\displaystyle{ sin^4(\alpha)\cdot cos^4(\alpha) = \frac{ 3 - 4\cdot cos(4\alpha) + cos(8\alpha) }{128} } sin5(α)⋅cos5(α)=10⋅sin(2α)−5⋅sin(6α)+sin(10α)512\displaystyle{ sin^5(\alpha)\cdot cos^5(\alpha) = \frac{ 10\cdot sin(2\alpha) - 5\cdot sin(6\alpha) + sin(10\alpha) }{512} }

Формулы понижения степени

sinn(α)=Cn2n2n+12n−1∑k=0n2−1(−1)n2−kCkncos((n−2k)α)\displaystyle{ sin^n(\alpha) = \frac{C_{\frac{n}{2}}^{n}}{2^n} + \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{\frac{n}{2}-k} C_{k}^{n}cos((n-2k)\alpha) } cosn(α)=Cn2n2n+12n−1∑k=0n2−1Ckncos((n−2k)α)\displaystyle{ cos^n(\alpha) = \frac{C_{\frac{n}{2}}^{n}}{2^n} + \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n}{2}-1} C_{k}^{n}cos((n-2k)\alpha) } sinn(α)=12n−1∑k=0n−12(−1)n−12−kCknsin((n−2k)α)\displaystyle{ sin^n(\alpha) = \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{\frac{n-1}{2}-k} C_{k}^{n}sin((n-2k)\alpha) } cosn(α)=12n−1∑k=0n−12Ckncos((n−2k)α)\displaystyle{ cos^n(\alpha) = \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n-1}{2}} C_{k}^{n}cos((n-2k)\alpha) }

Универсальная тригонометрическая подстановка

sin(α)=2⋅td(α2)1+tg2(α2)\displaystyle{ sin(\alpha) = \frac{2\cdot td \left( \frac{\alpha}{2} \right)}{1 + tg^2 \left( \frac{\alpha}{2} \right)} } cos(α)=1−tg2(α2)1+tg2(α2)\displaystyle{ cos(\alpha) = \frac{ 1 - tg^2 \left( \frac{\alpha}{2} \right) }{1 + tg^2 \left( \frac{\alpha}{2} \right)} } tg(α)=2⋅td(α2)1−tg2(α2)\displaystyle{ tg(\alpha) = \frac{2\cdot td \left( \frac{\alpha}{2} \right)}{1 - tg^2 \left( \frac{\alpha}{2} \right)} } ctg(α)=1−tg2(α2)2⋅td(α2)\displaystyle{ ctg(\alpha) = \frac{1 - tg^2 \left( \frac{\alpha}{2} \right)}{2\cdot td \left( \frac{\alpha}{2} \right)} }

Значения тригонометрических функций

α 0 π6\displaystyle{ \frac{\pi}{6} } π4\displaystyle{ \frac{\pi}{4} } π3\displaystyle{ \frac{\pi}{3} } π2\displaystyle{ \frac{\pi}{2} } 2π3\displaystyle{ \frac{2\pi}{3} } 3π4\displaystyle{ \frac{3\pi}{4} } 5π6\displaystyle{ \frac{5\pi}{6} } π\displaystyle{ \pi } 7π6\displaystyle{ \frac{7\pi}{6} } 5π4\displaystyle{ \frac{5\pi}{4} } 4π3\displaystyle{ \frac{4\pi}{3} } 3π2\displaystyle{ \frac{3\pi}{2} } 5π3\displaystyle{ \frac{5\pi}{3} } 7π4\displaystyle{ \frac{7\pi}{4} } 11π6\displaystyle{ \frac{11\pi}{6} } 2π\displaystyle{ 2\pi }
α° 0° 30° 45° 60° 90° 120° 135° 150° 180° 210° 225° 240° 270° 300° 315° 330° 360°
sin α 0 12\displaystyle{ \frac{1}{2} } 22\displaystyle{ \frac{\sqrt{2}}{2} } 32\displaystyle{ \frac{\sqrt{3}}{2} } 1 32\displaystyle{ \frac{\sqrt{3}}{2} } 22\displaystyle{ \frac{\sqrt{2}}{2} } 12\displaystyle{ \frac{1}{2} } 0 −12\displaystyle{ -\frac{1}{2} } −22\displaystyle{ -\frac{\sqrt{2}}{2} } −32\displaystyle{ -\frac{\sqrt{3}}{2} } −1 −32\displaystyle{ -\frac{\sqrt{3}}{2} } −22\displaystyle{ -\frac{\sqrt{2}}{2} } −12\displaystyle{ -\frac{1}{2} } 0
cos α 1 32\displaystyle{ \frac{\sqrt{3}}{2} } 22\displaystyle{ \frac{\sqrt{2}}{2} } 12\displaystyle{ \frac{1}{2} } 0 −12\displaystyle{ -\frac{1}{2} } −22\displaystyle{ -\frac{\sqrt{2}}{2} } −32\displaystyle{ -\frac{\sqrt{3}}{2} } −1 −32\displaystyle{ -\frac{\sqrt{3}}{2} } −22\displaystyle{ -\frac{\sqrt{2}}{2} } −12\displaystyle{ -\frac{1}{2} } 0 12\displaystyle{ \frac{1}{2} } 22\displaystyle{ \frac{\sqrt{2}}{2} } 32\displaystyle{ \frac{\sqrt{3}}{2} } 1
tg α 0 13\displaystyle{ \frac{1}{\sqrt{3}} } 1 3\displaystyle{ \sqrt{3} } − −3\displaystyle{ - \sqrt{3} } −1 −13\displaystyle{ -\frac{1}{\sqrt{3}} } 0 13\displaystyle{ \frac{1}{\sqrt{3}} } 1 3\displaystyle{ \sqrt{3} } − −3\displaystyle{ - \sqrt{3} } −1 −13\displaystyle{ - \frac{1}{\sqrt{3}} } 0
ctg α − 3\displaystyle{ \sqrt{3} } 1 13\displaystyle{ \frac{1}{\sqrt{3}} } 0 −13\displaystyle{ - \frac{1}{\sqrt{3}} } −1 −3\displaystyle{ - \sqrt{3} } − 3\displaystyle{ \sqrt{3} } 1 13\displaystyle{ \frac{1}{\sqrt{3}} } 0 −13\displaystyle{ - \frac{1}{\sqrt{3}} } −1 −3\displaystyle{ - \sqrt{3} } −
Алгебра Математика Формулы
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