Pythagorean Identities
{ cos 2 t + sin 2 t = 1 1 + tan 2 t = sec 2 t 1 + cot 2 t = csc 2 t \begin{cases}{\cos }^{2}t+{\sin }^{2}t=1\\ 1+{\tan }^{2}t={\sec }^{2}t\\ 1+{\cot }^{2}t={\csc }^{2}t\end{cases} ⎩ ⎨ ⎧ cos 2 t + sin 2 t = 1 1 + tan 2 t = sec 2 t 1 + cot 2 t = csc 2 t
Even-Odd Identities
{ cos ( − t ) = cos t sec ( − t ) = sec t sin ( − t ) = − sin t tan ( − t ) = − tan t csc ( − t ) = − csc t cot ( − t ) = − cot t \begin{cases}\cos \left(-t\right)=\cos t\hfill \\ \sec \left(-t\right)=\sec t\hfill \\ \sin \left(-t\right)=-\sin t\hfill \\ \tan \left(-t\right)=-\tan t\hfill \\ \csc \left(-t\right)=-\csc t\hfill \\ \cot \left(-t\right)=-\cot t\hfill \end{cases} ⎩ ⎨ ⎧ cos ( − t ) = cos t sec ( − t ) = sec t sin ( − t ) = − sin t tan ( − t ) = − tan t csc ( − t ) = − csc t cot ( − t ) = − cot t
Cofunction Identities
{ cos t = sin ( π 2 − t ) sin t = cos ( π 2 − t ) tan t = cot ( π 2 − t ) cot t = tan ( π 2 − t ) sec t = csc ( π 2 − t ) csc t = sec ( π 2 − t ) \begin{cases}\cos t=\sin \left(\frac{\pi }{2}-t\right)\hfill \\ \sin t=\cos \left(\frac{\pi }{2}-t\right)\hfill \\ \tan t=\cot \left(\frac{\pi }{2}-t\right)\hfill \\ \cot t=\tan \left(\frac{\pi }{2}-t\right)\hfill \\ \sec t=\csc \left(\frac{\pi }{2}-t\right)\hfill \\ \csc t=\sec \left(\frac{\pi }{2}-t\right)\hfill \end{cases} ⎩ ⎨ ⎧ cos t = sin ( 2 π − t ) sin t = cos ( 2 π − t ) tan t = cot ( 2 π − t ) cot t = tan ( 2 π − t ) sec t = csc ( 2 π − t ) csc t = sec ( 2 π − t )
Fundamental Identities
{ tan t = sin t cos t sec t = 1 cos t csc t = 1 sin t cot t = 1 tan t = cos t sin t \begin{cases}\tan t=\frac{\sin t}{\cos t}\hfill \\ \sec t=\frac{1}{\cos t}\hfill \\ \csc t=\frac{1}{\sin t}\hfill \\ \text{cot}t=\frac{1}{\text{tan}t}=\frac{\text{cos}t}{\text{sin}t}\hfill \end{cases} ⎩ ⎨ ⎧ tan t = c o s t s i n t sec t = c o s t 1 csc t = s i n t 1 cot t = tan t 1 = sin t cos t
Sum and Difference Identities
{ cos ( α + β ) = cos α cos β − sin α sin β cos ( α − β ) = cos α cos β + sin α sin β sin ( α + β ) = sin α cos β + cos α sin β sin ( α − β ) = sin α cos β − cos α sin β tan ( α + β ) = tan α + tan β 1 − tan α tan β tan ( α − β ) = tan α − tan β 1 + tan α tan β \begin{cases}\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta \hfill \\ \cos \left(\alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \hfill \\ \sin \left(\alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \hfill \\ \sin \left(\alpha -\beta \right)=\sin \alpha \cos \beta -\cos \alpha \sin \beta \hfill \\ \tan \left(\alpha +\beta \right)=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }\hfill \\ \tan \left(\alpha -\beta \right)=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta }\hfill \end{cases} ⎩ ⎨ ⎧ cos ( α + β ) = cos α cos β − sin α sin β cos ( α − β ) = cos α cos β + sin α sin β sin ( α + β ) = sin α cos β + cos α sin β sin ( α − β ) = sin α cos β − cos α sin β tan ( α + β ) = 1 − t a n α t a n β t a n α + t a n β tan ( α − β ) = 1 + t a n α t a n β t a n α − t a n β
Double-Angle Formulas
{ sin ( 2 θ ) = 2 sin θ cos θ cos ( 2 θ ) = cos 2 θ − sin 2 θ cos ( 2 θ ) = 1 − 2 sin 2 θ cos ( 2 θ ) = 2 cos 2 θ − 1 tan ( 2 θ ) = 2 tan θ 1 − tan 2 θ \begin{cases}\sin \left(2\theta \right)=2\sin \theta \cos \theta \hfill \\ \cos \left(2\theta \right)={\cos }^{2}\theta -{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=1 - 2{\sin }^{2}\theta \hfill \\ \cos \left(2\theta \right)=2{\cos }^{2}\theta -1\hfill \\ \tan \left(2\theta \right)=\frac{2\tan \theta }{1-{\tan }^{2}\theta }\hfill \end{cases} ⎩ ⎨ ⎧ sin ( 2 θ ) = 2 sin θ cos θ cos ( 2 θ ) = cos 2 θ − sin 2 θ cos ( 2 θ ) = 1 − 2 sin 2 θ cos ( 2 θ ) = 2 cos 2 θ − 1 tan ( 2 θ ) = 1 − t a n 2 θ 2 t a n θ
Half-Angle Formulas
{ sin α 2 = ± 1 − cos α 2 cos α 2 = ± 1 + cos α 2 tan α 2 = ± 1 − cos α 1 + cos α tan α 2 = sin α 1 + cos α tan α 2 = 1 − cos α sin α \begin{cases}\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\hfill \\ \cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\hfill \\ \tan \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{1+\cos \alpha }}\hfill \\ \tan \frac{\alpha }{2}=\frac{\sin \alpha }{1+\cos \alpha }\hfill \\ \tan \frac{\alpha }{2}=\frac{1-\cos \alpha }{\sin \alpha }\hfill \end{cases} ⎩ ⎨ ⎧ sin 2 α = ± 2 1 − c o s α cos 2 α = ± 2 1 + c o s α tan 2 α = ± 1 + c o s α 1 − c o s α tan 2 α = 1 + c o s α s i n α tan 2 α = s i n α 1 − c o s α
Reduction Formulas
{ sin 2 θ = 1 − cos ( 2 θ ) 2 cos 2 θ = 1 + cos ( 2 θ ) 2 tan 2 θ = 1 − cos ( 2 θ ) 1 + cos ( 2 θ ) \begin{cases}{\sin }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{2}\\ {\cos }^{2}\theta =\frac{1+\cos \left(2\theta \right)}{2}\\ {\tan }^{2}\theta =\frac{1-\cos \left(2\theta \right)}{1+\cos \left(2\theta \right)}\end{cases} ⎩ ⎨ ⎧ sin 2 θ = 2 1 − c o s ( 2 θ ) cos 2 θ = 2 1 + c o s ( 2 θ ) tan 2 θ = 1 + c o s ( 2 θ ) 1 − c o s ( 2 θ )
Product-to-Sum Formulas
{ cos α cos β = 1 2 [ cos ( α − β ) + cos ( α + β ) ] sin α cos β = 1 2 [ sin ( α + β ) + sin ( α − β ) ] sin α sin β = 1 2 [ cos ( α − β ) − cos ( α + β ) ] cos α sin β = 1 2 [ sin ( α + β ) − sin ( α − β ) ] \begin{cases}\cos \alpha \cos \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)+\cos \left(\alpha +\beta \right)\right]\hfill \\ \sin \alpha \cos \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)+\sin \left(\alpha -\beta \right)\right]\hfill \\ \sin \alpha \sin \beta =\frac{1}{2}\left[\cos \left(\alpha -\beta \right)-\cos \left(\alpha +\beta \right)\right]\hfill \\ \cos \alpha \sin \beta =\frac{1}{2}\left[\sin \left(\alpha +\beta \right)-\sin \left(\alpha -\beta \right)\right]\hfill \end{cases} ⎩ ⎨ ⎧ cos α cos β = 2 1 [ cos ( α − β ) + cos ( α + β ) ] sin α cos β = 2 1 [ sin ( α + β ) + sin ( α − β ) ] sin α sin β = 2 1 [ cos ( α − β ) − cos ( α + β ) ] cos α sin β = 2 1 [ sin ( α + β ) − sin ( α − β ) ]
Sum-to-Product Formulas
{ sin α + sin β = 2 sin ( α + β 2 ) cos ( α − β 2 ) sin α − sin β = 2 sin ( α − β 2 ) cos ( α + β 2 ) cos α − cos β = − 2 sin ( α + β 2 ) sin ( α − β 2 ) cos α + cos β = 2 cos ( α + β 2 ) cos ( α − β 2 ) \begin{cases}\sin \alpha +\sin \beta =2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \sin \alpha -\sin \beta =2\sin \left(\frac{\alpha -\beta }{2}\right)\cos \left(\frac{\alpha +\beta }{2}\right)\hfill \\ \cos \alpha -\cos \beta =-2\sin \left(\frac{\alpha +\beta }{2}\right)\sin \left(\frac{\alpha -\beta }{2}\right)\hfill \\ \cos \alpha +\cos \beta =2\cos \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)\hfill \end{cases} ⎩ ⎨ ⎧ sin α + sin β = 2 sin ( 2 α + β ) cos ( 2 α − β ) sin α − sin β = 2 sin ( 2 α − β ) cos ( 2 α + β ) cos α − cos β = − 2 sin ( 2 α + β ) sin ( 2 α − β ) cos α + cos β = 2 cos ( 2 α + β ) cos ( 2 α − β )
Law of Sines
{ sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ \begin{cases}\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}\hfill \\ \frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }\hfill \end{cases} { a s i n α = b s i n β = c s i n γ s i n α a = s i n β b = s i n γ c
Law of Cosines
{ a 2 = b 2 + c 2 − 2 b c cos α b 2 = a 2 + c 2 − 2 a c cos β c 2 = a 2 + b 2 − 2 a b cos γ \begin{cases}{a}^{2}={b}^{2}+{c}^{2}-2bc\cos \alpha \hfill \\ {b}^{2}={a}^{2}+{c}^{2}-2ac\cos \beta \hfill \\ {c}^{2}={a}^{2}+{b}^{2}-2ab\text{cos}\gamma \hfill \end{cases} ⎩ ⎨ ⎧ a 2 = b 2 + c 2 − 2 b c cos α b 2 = a 2 + c 2 − 2 a c cos β c 2 = a 2 + b 2 − 2 ab cos γ