Fórmulas Trigonométricas

$$ \mathbf{\hbox{(Fórmulas Fundamentales de la Trigonometría)}} \left\{\begin{array}{l} sen^2(\alpha) + cos^2(\alpha) = 1 \\ tan^2(\alpha) + 1 = cosec^2(\alpha) \\ 1 + cotan^2(\alpha) = sec^2(\alpha) \\ \end{array}\right.\\ $$ $$ \mathbf{\hbox{(Suma de ángulos)}} \left\{\begin{array}{l} sen(\alpha + \beta) = sen(\alpha) \cdot cos(\beta) + cos(\alpha) \cdot sen(\beta) \\ cos(\alpha + \beta) = cos(\alpha) \cdot cos(\beta) - sen(\alpha) \cdot sen(\beta) \\ tan(\alpha + \beta) = \displaystyle{\frac{tan(\alpha) + tan(\beta)}{1 - tan(\alpha) \cdot tan(\beta)}} \end{array}\right.\\ $$ $$ \mathbf{\hbox{(Diferencia de ángulos)}} \left\{\begin{array}{l} sen(\alpha - \beta) = sen(\alpha) \cdot cos(\beta) - cos(\alpha) \cdot sen(\beta) \\ cos(\alpha - \beta) = cos(\alpha) \cdot cos(\beta) + sen(\alpha) \cdot sen(\beta) \\ tan(\alpha - \beta) = \displaystyle{\frac{tan(\alpha) - tan(\beta)}{1 + tan(\alpha) \cdot tan(\beta)}} \end{array}\right.\\ $$ $$ \mathbf{\hbox{(Ángulo doble)}} \left\{\begin{array}{l} sen(2\alpha) = 2sen(\alpha) \cdot cos(\alpha)\\ cos(2\alpha) = cos^2(\alpha) - sen^2(\alpha) = 2cos^2(\alpha) - 1\\ tan(2\alpha) = \displaystyle{\frac{2tan(\alpha)}{1 + tan^2(\alpha)}} \end{array}\right.\\ $$ $$ \mathbf{\hbox{(Ángulo mitad)}} \left\{\begin{array}{l} sen(\displaystyle{\frac{\alpha}{2}}) = \sqrt[]{\displaystyle{\frac{1 - cos(\alpha)}{2}}} \\ cos(\displaystyle{\frac{\alpha}{2}}) = \sqrt[]{\displaystyle{\frac{1 + cos(\alpha)}{2}}} \\ tan(\displaystyle{\frac{\alpha}{2}}) = \sqrt[]{\displaystyle{\frac{1 - cos(\alpha)}{1 + cos(\alpha)}}} \end{array}\right. $$

Resolución de triángulos

Dado un triángulo de lados \(a\), \(b\) y \(c\) y ángulos \(\hat{A}\), \(\hat{B}\) y \(\hat{C}\):

$$ \mathbf{\hbox{ (Teorema del seno) }} \frac{a}{sen(\hat{A})}=\frac{b}{sen(\hat{B})}=\frac{c}{sen(\hat{C})}=2R $$ $$ \mathbf{\hbox{ (Teorema del coseno) }} \left\{\begin{array}{l} a^2 = b^2 + c^2 - 2bc \cdot cos(\hat{A})\\ b^2 = a^2 + c^2 - 2ac \cdot cos(\hat{B})\\ c^2 = a^2 + b^2 - 2ab \cdot cos(\hat{C})\\ \end{array}\right. $$ $$ \mathbf{\hbox{ (Transformación de sumas a productos) }} \left\{\begin{array}{l} sen(\alpha) + sen(\beta) = 2sen(\displaystyle{\frac{\alpha + \beta}{2}}) cos(\displaystyle{\frac{\alpha - \beta}{2}})\\ sen(\alpha) - sen(\beta) = 2cos(\displaystyle{\frac{\alpha + \beta}{2}}) sen(\displaystyle{\frac{\alpha - \beta}{2}})\\ cos(\alpha) + cos(\beta) = 2cos(\displaystyle{\frac{\alpha + \beta}{2}}) cos(\displaystyle{\frac{\alpha - \beta}{2}})\\ cos(\alpha) - cos(\beta) = -2sen(\displaystyle{\frac{\alpha + \beta}{2}}) sen(\displaystyle{\frac{\alpha - \beta}{2}})\\ tan(\alpha) + tan(\beta) = \displaystyle{\frac{sen(\alpha + \beta)}{cos(\alpha)cos(\beta)}}\\ tan(\alpha) - tan(\beta) = \displaystyle{\frac{sen(\alpha - \beta)}{cos(\alpha)cos(\beta)}}\\ \end{array}\right.\\ $$ $$ \mathbf{\hbox{ (Transformación de productos a sumas) }} \left\{\begin{array}{l} sen(\alpha)sen(\beta) = \displaystyle{\frac{1}{2}} \left [cos(\alpha - \beta) - cos(\alpha + \beta) \right ]\\ sen(\alpha)cos(\beta) = \displaystyle{\frac{1}{2}} \left [sen(\alpha + \beta) + sen(\alpha - \beta) \right ]\\ cos(\alpha)cos(\beta) = \displaystyle{\frac{1}{2}} \left [cos(\alpha + \beta) + cos(\alpha - \beta) \right ]\\ \end{array}\right. $$

Área del triángulo (S)

$$ \mathbf{\hbox{ (Fórmula de Herón) }} S = \sqrt{p(p-a)(p-b)(p-c)}, \hbox{ donde } p=\frac{a+b+c}{2} $$ $$ S = \frac{1}{2}bc \cdot sen(\hat{A}) = \frac{1}{2}ac \cdot sen(\hat{B}) = \frac{1}{2}ab \cdot sen(\hat{C}) $$

Más fórmulas trigonométricas (Clica para mostrar)