Question

Sendo cossec x = -5/3, calcule o valor de cos (2x)

Answer 1

Temos o seguinte:

$\mathtt{csc(x) = - \frac{5}{3} \to \frac{1}{sin(x)} = -\frac{5}{3} \to sin(x) = - \frac{3}{5} }$

Assim:

$\mathtt{sin^2(x) + cos^2(x) = 1 \to cos^2(x) = 1 - sin^2(x)} \\ \\ \mathtt{cos^2(x) = 1 - ( -\frac{3}{5} )^2 \to cos^2(x) = 1 - \frac{9}{25} \to cos^2(x) = \frac{16}{25} } \\ \\ \mathtt{cos(x) = \sqrt{ \frac{16}{25} } \to cos(x) = \frac{4}{5} }$

Logo temos:

$\mathtt{cos(2x) = cos(x+x) = cos(x)*cos(x) - sin(x)*sin(x) } \\ \\ \mathtt{cos(x)*cos(x) - sin(x)*sin(x) = cos^2(x) - sin^2(x)}$

Logo:

$\mathtt{cos^2(x) - sin^2(x) = ( \frac{4}{5} )^2 - ( -\frac{3}{5} )^2 = \frac{16}{25} -\frac{9}{25} = \frac{7}{25} }$