Relações Trigonométricas Usuais

$$\sin^2 x + \cos^2 x = 1 \qquad \cos^2 x = {{1+\cos 2x}\over 2}\qquad \sin^2 x = {{1-\cos 2x}\over 2}$$

$$\sin x = {{e^{jx} - e^{-jx}}\over 2j} \qquad \cos x = {{e^{j... ...quad \tan x = {{e^{jx} - e^{-jx}}\over {j(e^{jx} + e^{-jx})} }$$

Adição

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

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

$$\tan (a+b) = {{\tan a + \tan b}\over {1-\tan a \tan b}} \qquad \tan (a-b) = {{\tan a - \tan b}\over {1+\tan a \tan b}}$$

Multiplicação: com $\tan a = t$

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

$$\cos (2a) = \cos^2 a - \sin^2 a = 2\cos^2 a -1 = 1-2\sin^2 a = {{1-t^2}\over {1+t^2}}$$

$$\tan (2a) = {{2\tan a}\over {1-\tan^2 a}} = {{2t}\over {1-t^2}}$$

$$\cos a \cos b = {1\over 2} [\cos(a+b) + \cos (a-b)] \qquad \sin a \sin b = {1\over 2} [\cos(a-b) - \cos(a+b)]$$

$$\sin a \cos b = {1\over 2} [\sin(a+b) + \sin(a-b)]$$

$$\cos p + \cos q = 2 \cos {{p+q}\over 2} \cos {{p-q}\over 2}\qquad \cos p - \cos q = -2 \sin {{p+q}\over 2} \sin {{p-q}\over 2}$$

$$\sin p + \sin q = 2 \sin {{p+q}\over 2} \cos {{p-q}\over 2}\qquad \sin p - \sin q = 2 \sin {{p-q}\over 2} \cos {{p+q}\over 2}$$

$$\tan p + \tan q = {{\sin(p+q)}\over {\cos p \cos q}}\qquad \tan p - \tan q = {{\sin(p-q)}\over {\cos p \cos q}}$$

Trigonometria Hiperbólica

$$\cosh x + \sinh x = \exp(x) \qquad \cosh x - \sinh x = \exp(-x) \qquad \cosh^2 x - \sinh^2 x = 1$$

$$\sinh(a+b) = \sinh a \cosh b + \sinh b \cosh a \qquad \sinh (a-b) = \sinh a \cosh b - \sinh b \cosh a$$

$$\cosh (a+b) = \cosh a \cosh b + \sinh a \sinh b \qquad \cosh(a-b) = \cosh a \cosh b - \sinh a \sinh b$$

$$\sinh 2a = 2\sinh a \cosh a \qquad \cosh 2a = \cosh^2 a \sinh^2 a = 1+2\sin^2 a = 2\cosh^2 a -1$$

$$\cosh^2 a = {{1+\cosh 2a}\over 2} \qquad \sinh^2 a = {{\cosh... ...\over 2} \qquad \tanh^2 a = {{\cosh 2a -1}\over {\cosh 2a +1}}$$

$$\tanh 2a = {{2\tanh a}\over {1+\tanh^2 a}}$$

$$\sinh x = {{e^{x} - e^{-x}}\over 2}\qquad \cosh = {{e^{x} + ... ...ver 2}\qquad \tanh = {{e^{x} - e^{-x}}\over {e^{x} + e^{-x}} }$$

$$\cosh jx = \cos x \qquad \sinh jx = j \sin x\qquad \tanh jx = j\tan x$$