Question

Calcule cosx , sabendo que 2senx + cosx = 2

por favor to desesperado

Answer 1

Usando a relação fundamental da trigonometria :

$\text{Sen}^2(\text x) + \text{Cos}^2(\text x) = 1$

sabemos que :

$\text{Sen(x)} = \sqrt{1-\text{Cos}^2(\text x)}$

Então basta substituir isso na equação e resolver para o Cosseno.

$2\text{Sen(x)}+\text{Cos(x)} = 2$

$2\sqrt{1-\text{Cos}^2(\text x)} +\text{Cos(x)} = 2$

$2\sqrt{1-\text{Cos}^2(\text x)} = 2 -\text{Cos(x)}$

Elevando ao quadrado dos dois lados :

$4[1-\text{Cos}^2(\text x)] = 4 -4\text{Cos(x)}+\text{Cos}^2(\text x)$

$4 -4\text{Cos}^2(\text x)] = 4 -4\text{Cos(x)}+\text{Cos}^2(\text x)$

$5\text{Cos}^2(\text x)-4\text{Cos(x)} = 0$

$\text{Cos(x)}[5.\text{Cos}(\text x)-4] = 0$

temos :

$\text{Cos(x)} = 0$

ou

$5.\text{Cos(x)} -4=0 \to {\displaystyle \text{Cos(x)} = \frac{4}{5}}$

Portanto :

$\huge\boxed{{\displaystyle \text{Cos(x)} = \frac{4}{5}} \ \ \text{ou} \ \ {\displaystyle \text{Cos(x)} =0} }\checkmark$