To find [tex]\(\cos 2x\)[/tex] given that [tex]\(\sin x = \frac{2}{3}\)[/tex], we can follow these steps:
1. Use the Pythagorean identity to find [tex]\(\cos x\)[/tex].
The Pythagorean identity is:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Given [tex]\(\sin x = \frac{2}{3}\)[/tex], first compute [tex]\(\sin^2 x\)[/tex]:
[tex]\[
\sin^2 x = \left( \frac{2}{3} \right)^2 = \frac{4}{9}
\][/tex]
Substitute [tex]\(\sin^2 x\)[/tex] into the Pythagorean identity and solve for [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\cos^2 x = 1 - \sin^2 x = 1 - \frac{4}{9}
\][/tex]
Simplify the right-hand side:
[tex]\[
\cos^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}
\][/tex]
2. Compute [tex]\(\cos x\)[/tex]:
[tex]\[
\cos x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
\][/tex]
Since the value of [tex]\(\cos x\)[/tex] should be non-negative in this context, we take the positive square root:
[tex]\[
\cos x \approx 0.745356
\][/tex]
3. Use the double-angle formula for cosine to find [tex]\(\cos 2x\)[/tex]:
The double-angle formula for cosine is:
[tex]\[
\cos 2x = 2 \cos^2 x - 1
\][/tex]
Substitute [tex]\(\cos^2 x = \frac{5}{9}\)[/tex] into the formula:
[tex]\[
\cos 2x = 2 \left( \frac{5}{9} \right) - 1
\][/tex]
Simplify the expression:
[tex]\[
\cos 2x = \frac{10}{9} - 1 = \frac{10}{9} - \frac{9}{9} = \frac{1}{9}
\][/tex]
So, the value of [tex]\(\cos 2x\)[/tex] is:
[tex]\[
\cos 2x \approx 0.1111
\][/tex]
