Certainly! Let's solve the problem step-by-step:
Step 1: Understanding the Given Information
We are given that [tex]\(\sin x = \frac{2}{3}\)[/tex]. We are asked to find the value of [tex]\(\cos 2x\)[/tex].
Step 2: Using the Pythagorean Identity
Recall the Pythagorean identity for sine and cosine:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Given:
[tex]\[
\sin x = \frac{2}{3}
\][/tex]
First, we find [tex]\(\sin^2 x\)[/tex]:
[tex]\[
\sin^2 x = \left(\frac{2}{3}\right)^2 = \frac{4}{9}
\][/tex]
Now, using the Pythagorean identity:
[tex]\[
\cos^2 x = 1 - \sin^2 x = 1 - \frac{4}{9}
\][/tex]
Let’s find the value of [tex]\(1 - \frac{4}{9}\)[/tex]:
[tex]\[
\cos^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}
\][/tex]
Therefore:
[tex]\[
\cos^2 x = \frac{5}{9}
\][/tex]
And then:
[tex]\[
\cos x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
\][/tex]
Step 3: Using the Double-Angle Formula
The double-angle formula for cosine is:
[tex]\[
\cos 2x = 2\cos^2 x - 1
\][/tex]
We already found that:
[tex]\[
\cos^2 x = \frac{5}{9}
\][/tex]
Substitute [tex]\(\cos^2 x\)[/tex] into the double-angle formula:
[tex]\[
\cos 2x = 2 \left(\frac{5}{9}\right) - 1
\][/tex]
Let’s simplify:
[tex]\[
\cos 2x = \frac{10}{9} - 1 = \frac{10}{9} - \frac{9}{9} = \frac{1}{9}
\][/tex]
Therefore, the value of [tex]\(\cos 2x\)[/tex] is:
[tex]\[
\cos 2x = \frac{1}{9}
\][/tex]
