Sure, let’s solve the given question step by step.
Given:
[tex]\[
\sin(x) = -\frac{3}{5}
\][/tex]
and
[tex]\[
\cos(x) < 0
\][/tex]
First, we'll use the Pythagorean identity to find [tex]\(\cos(x)\)[/tex]. The Pythagorean identity is:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
Substitute the given [tex]\(\sin(x)\)[/tex] value into the identity:
[tex]\[
\left(-\frac{3}{5}\right)^2 + \cos^2(x) = 1
\][/tex]
[tex]\[
\frac{9}{25} + \cos^2(x) = 1
\][/tex]
Solving for [tex]\(\cos^2(x)\)[/tex]:
[tex]\[
\cos^2(x) = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2(x) = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2(x) = \frac{16}{25}
\][/tex]
Since [tex]\(\cos(x) < 0\)[/tex], we take the negative square root:
[tex]\[
\cos(x) = -\sqrt{\frac{16}{25}} = -\frac{4}{5}
\][/tex]
Next, we'll use the double-angle formula for cosine:
[tex]\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\][/tex]
Substituting the known values of [tex]\(\cos(x)\)[/tex] and [tex]\(\sin(x)\)[/tex]:
[tex]\[
\cos(2x) = \left(-\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2
\][/tex]
[tex]\[
\cos(2x) = \frac{16}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\cos(2x) = \frac{16}{25} - \frac{9}{25} = \frac{7}{25}
\][/tex]
Therefore, the value of [tex]\(\cos(2x)\)[/tex] is:
[tex]\[
\boxed{\frac{7}{25}}
\][/tex]
