Trigonometry

Sine and cosine

$sin(a)^{2}+cos(a)^{2} = 1$

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Tangent

$tg(a) = \frac{sin(a)}{cos(a)}$

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Cotangent

$ctg(a) = \frac{cos(a)}{sin(a)}$

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Product of tangent and cotangent

$tg(a)\cdot ctg(a) = 1$

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Tangent and cosine

$1+tg(a)^{2} = \frac{1}{cos(a)^{2}}$

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Cotangent and sine

$1+ctg(a)^{2} = \frac{1}{sin(a)^{2}}$

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Sine of sum of angles

$sin(a+b) = sin(a)\cdot cos(b)+cos(a)\cdot sin(b)$

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Sine of difference of angles

$sin(a-b) = sin(a)\cdot cos(b)-cos(a)\cdot sin(b)$

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Cosine of sum of angles

$cos(a+b) = cos(a)\cdot cos(b)-sin(a)\cdot sin(b)$

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Cosine of difference of angles

$cos(a-b) = cos(a)\cdot cos(b)+sin(a)\cdot sin(b)$

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Tangent of sum of angles

$tg(a+b) = \frac{tg(a)+tg(b)}{1-tg(a)\cdot tg(b)}$

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Tangent of difference of angles

$tg(a-b) = \frac{tg(a)-tg(b)}{1+tg(a)\cdot tg(b)}$

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Sine of double angle

$sin(2\cdot a) = 2\cdot sin(a)\cdot cos(a)$

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Cosine of double angle

$cos(2\cdot a) = cos(a)^{2}-sin(a)^{2}$

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Cosine of double angle

$cos(2\cdot a) = 1-2\cdot sin(a)^{2}$

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Cosine of double angle

$cos(2\cdot a) = 2\cdot cos(a)^{2}-1$

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Tangent of double angle

$tg(2\cdot a) = \frac{2\cdot tg(a)}{1-tg(a)^{2}}$

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Cotangent of double angle

$ctg(2\cdot a) = \frac{1-tg(a)^{2}}{2\cdot tg(a)}$

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Cotangent of double angle

$ctg(2\cdot a) = \frac{ctg(a)-tg(a)}{2}$

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Sum of sines (sum to product)

$sin(a)+sin(b) = 2\cdot sin(\frac{a+b}{2})\cdot cos(\frac{a-b}{2})$

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Difference of sines (difference to product)

$sin(a)-sin(b) = 2\cdot cos(\frac{a+b}{2})\cdot sin(\frac{a-b}{2})$

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Sum of cosines (sum to product)

$cos(a)+cos(b) = 2\cdot cos(\frac{a+b}{2})\cdot cos(\frac{a-b}{2})$

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Difference of cosines (difference to product)

$cos(a)-cos(b) = -2\cdot sin(\frac{a+b}{2})\cdot sin(\frac{a-b}{2})$

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Product of sine and cosine

$sin(a)\cdot cos(b) = \frac{1}{2}\cdot (sin(a-b)+sin(a+b))$

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Product of sines

$sin(a)\cdot sin(b) = \frac{1}{2}\cdot (cos(a-b)-cos(a+b))$

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Product of cosines

$cos(a)\cdot cos(b) = \frac{1}{2}\cdot (cos(a-b)+cos(a+b))$

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Power reduction for sine

$sin(a)^{2} = \frac{1-cos(2\cdot a)}{2}$

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Power reduction for cosine

$cos(a)^{2} = \frac{1+cos(2\cdot a)}{2}$

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