To find the value of [tex]\(\cos 2x\)[/tex] given that [tex]\(\cos x = \frac{-2}{5}\)[/tex], we can use the double-angle formula for cosine.
The double-angle formula for cosine is given by:
[tex]\[
\cos 2x = 2\cos^2 x - 1
\][/tex]
Given:
[tex]\[
\cos x = \frac{-2}{5}
\][/tex]
First, we need to find [tex]\(\cos^2 x\)[/tex]. This can be computed by squaring [tex]\(\cos x\)[/tex]:
[tex]\[
\cos^2 x = \left( \frac{-2}{5} \right)^2 = \frac{4}{25}
\][/tex]
Next, we substitute [tex]\(\cos^2 x\)[/tex] into the double-angle formula:
[tex]\[
\cos 2x = 2\cos^2 x - 1
\][/tex]
Substituting [tex]\(\cos^2 x\)[/tex] value:
[tex]\[
\cos 2x = 2 \left( \frac{4}{25} \right) - 1
\][/tex]
Now, simplify the expression:
[tex]\[
\cos 2x = 2 \cdot \frac{4}{25} - 1 = \frac{8}{25} - 1
\][/tex]
Convert 1 to a fraction with the same denominator:
[tex]\[
1 = \frac{25}{25}
\][/tex]
Then subtract:
[tex]\[
\cos 2x = \frac{8}{25} - \frac{25}{25} = \frac{8 - 25}{25} = \frac{-17}{25}
\][/tex]
Therefore, the value of [tex]\(\cos 2x\)[/tex] is:
[tex]\[
\cos 2x = \frac{-17}{25} \approx -0.68
\][/tex]
