Question

Se sen (a/2)=1/5, calcular cos a.

Answer 1

Usaremos uma das identidades para calcular o cosseno do arco duplo:

     •   $\mathsf{cos\,2x=cos^2\,x-sen^2\,x\qquad\quad (i)}$


combinada com a Relação Trigonométrica Fundamental:

     •   $\mathsf{cos^2\,x+sen^2\,x=1\qquad\quad (ii)}$

======

Usando  (i),  para  x = a/2,

     $\mathsf{cos\bigg(2\cdot \dfrac{a}{2}\bigg)=cos^2\bigg(\dfrac{a}{2}\bigg)-sen^2\bigg(\dfrac{a}{2}\bigg)}\\\\\\ \mathsf{cos\,a=cos^2\bigg(\dfrac{a}{2}\bigg)-sen^2\bigg(\dfrac{a}{2}\bigg)}$


Mas por  (ii), temos que  

     $\mathsf{cos^2\bigg(\dfrac{a}{2}\bigg)=1-sen^2\bigg(\dfrac{a}{2}\bigg)}$


Então, substituindo, ficamos com

     $\mathsf{cos\,a=1-sen^2\bigg(\dfrac{a}{2}\bigg)-sen^2\bigg(\dfrac{a}{2}\bigg)}\\\\\\ \mathsf{cos\,a=1-2\,sen^2\bigg(\dfrac{a}{2}\bigg)\qquad\quad }\textsf{mas }\mathsf{sen\bigg(\dfrac{a}{2}\bigg)=\dfrac{1}{5}},\textsf{~ent\~ao}\\\\\\ \mathsf{cos\,a=1-2\cdot \bigg(\dfrac{1}{5}\bigg)^{\!2}}\\\\\\ \mathsf{cos\,a=1-2\cdot \dfrac{1}{25}}\\\\\\ \mathsf{cos\,a=\dfrac{25}{25}-\dfrac{2}{25}}\\\\\\ \mathsf{cos\,a=\dfrac{25-2}{25}}$

     $\boxed{\begin{array}{c}\mathsf{cos\,a=\dfrac{23}{25}}\end{array}}$    <———    esta é a resposta.


Bons estudos! :-)