Answer :

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]

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