Trigonometry: Laws and Identities

$tan \theta = \frac{sin \theta }{cos \theta }$

$cot \theta = \frac{cos \theta }{sin \theta }$

$csc \theta = \frac{1}{sin \theta }$

$sec \theta = \frac{1}{cos \theta }$

$cot \theta = \frac{1}{tan \theta }$

$sin \theta = \frac{1}{csc \theta }$

$cos \theta = \frac{1}{sec \theta }$

$tan \theta = \frac{1}{cot \theta }$

$sin^2 \theta + cos^2 \theta = 1$

$tan^2 \theta + 1 = sec^2 \theta$

$cot^2 \theta + 1 = csc^2 \theta$

$sin( \theta + 2 \pi n) = sin \theta$

$cos( \theta + 2 \pi n) = cos \theta$

$tan( \theta +  \pi n) = tan \theta$

$csc( \theta + 2 \pi n) = csc \theta$

$sec( \theta + 2 \pi n) = sec \theta$

$cot( \theta +  \pi n) = cot \theta$

$sin(- \theta) = -sin \theta$

$cos(- \theta) = cos \theta$

$tan(- \theta) = -tan \theta$

$csc(- \theta) = -csc \theta$

$sec(- \theta) = sec \theta$

$cot(- \theta) = -cot \theta$

$sin(2 \theta) = 2sin \theta cos \theta$

$cos(2 \theta) = cos^2 \theta - sin^2 \theta$

$= 2cos^2 \theta - 1$

$= 1 - 2sin^2 \theta$

$tan(2 \theta) = \frac{2tan \theta }{1 - tan^2 \theta }$

$sin( \frac{ \theta }{2}) = \pm \sqrt[]{ \frac{1 - cos \theta }{2} }$

$cos( \frac{ \theta }{2}) = \pm \sqrt[]{ \frac{1 + cos \theta }{2} }$

$tan( \frac{ \theta }{2}) = \pm \sqrt[]{ \frac{1 - cos \theta }{1 + cos \theta } }$

$a^2 = b^2 + c^2 - 2bccos \alpha$

$b^2 = a^2 + c^2 - 2accos \beta$

$c^2 = a^2 + b^2 - 2abcos \gamma$

$sin \alpha sin \beta = \frac{1}{2}[cos( \alpha - \beta) - cos( \alpha + \beta)]$

$cos \alpha cos \beta = \frac{1}{2}[cos( \alpha - \beta) + cos( \alpha + \beta)]$

$sin \alpha cos \beta = \frac{1}{2}[sin( \alpha + \beta) + sin( \alpha - \beta)]$

$cos \alpha sin \beta = \frac{1}{2}[sin( \alpha + \beta) - sin( \alpha - \beta)]$

$sin \alpha + sin \beta = 2sin( \frac{ \alpha + \beta }{2})cos( \frac{ \alpha - \beta }{2})$

$sin \alpha - sin \beta = 2cos( \frac{ \alpha + \beta }{2})sin( \frac{ \alpha - \beta }{2})$

$cos \alpha + cos \beta = 2cos( \frac{ \alpha + \beta }{2})cos( \frac{ \alpha - \beta }{2})$

$cos \alpha - cos \beta = -2sin( \frac{ \alpha + \beta }{2})sin( \frac{ \alpha - \beta }{2})$

$\frac{a - b}{a + b} = \frac{tan[ \frac{1}{2}( \alpha - \beta)] }{tan[ \frac{1}{2}( \alpha + \beta)] }$

$\frac{b - c}{b+c} = \frac{tan[ \frac{1}{2}( \beta - \gamma)] }{tan[ \frac{1}{2}( \beta + \gamma)] }$

$\frac{a - c}{a+c} = \frac{tan[ \frac{1}{2}( \alpha - \gamma)] }{tan[ \frac{1}{2}( \alpha + \gamma)] }$

$sin( \alpha \pm \beta) = sin \alpha cos \beta \pm cos \alpha sin \beta$

$cos( \alpha \pm \beta) = cos \alpha cos \beta \mp sin \alpha sin \beta$

$tan( \alpha \pm \beta) = \frac{tan \alpha \pm tan \beta }{1 \mp tan \alpha tan \beta }$

$sin( \frac{ \pi }{2} - \theta) = cos \theta$

$csc( \frac{ \pi }{2} - \theta) = sec \theta$

$tan( \frac{ \pi }{2} - \theta) = cot \theta$

$cos( \frac{ \pi }{2} - \theta) = sin \theta$

$sec( \frac{ \pi }{2} - \theta) = csc \theta$

$cot( \frac{ \pi }{2} - \theta) = tan \theta$