If `"sin"x=2/(3)`, then find the value of Cos 3x.

Options

A. `-0.5797`
B. 0.5678
C. 0.6735
D. `-0.8765`

Answer

To find the value of \( \cos 3x \) given that \( \sin x = \frac{2}{3} \), we can follow these steps:

### Step 1: Find \( \cos x \)
We know the Pythagorean identity:
\[ \sin^2 x + \cos^2 x = 1 \]
Substituting \( \sin x = \frac{2}{3} \):
\[ \left(\frac{2}{3}\right)^2 + \cos^2 x = 1 \]
Calculating \( \left(\frac{2}{3}\right)^2 \):
\[ \frac{4}{9} + \cos^2 x = 1 \]
Now, isolate \( \cos^2 x \):
\[ \cos^2 x = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9} \]
Taking the square root gives:
\[ \cos x = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3} \]

### Step 2: Use the formula for \( \cos 3x \)
The formula for \( \cos 3x \) is:
\[ \cos 3x = 4 \cos^3 x - 3 \cos x \]
Substituting \( \cos x = \frac{\sqrt{5}}{3} \):
\[ \cos 3x = 4 \left(\frac{\sqrt{5}}{3}\right)^3 - 3 \left(\frac{\sqrt{5}}{3}\right) \]

### Step 3: Calculate \( \cos^3 x \)
Calculating \( \left(\frac{\sqrt{5}}{3}\right)^3 \):
\[ \left(\frac{\sqrt{5}}{3}\right)^3 = \frac{(\sqrt{5})^3}{3^3} = \frac{5\sqrt{5}}{27} \]
Now substitute back into the equation for \( \cos 3x \):
\[ \cos 3x = 4 \cdot \frac{5\sqrt{5}}{27} - 3 \cdot \frac{\sqrt{5}}{3} \]
This simplifies to:
\[ \cos 3x = \frac{20\sqrt{5}}{27} - \frac{9\sqrt{5}}{27} \]
Combining the terms:
\[ \cos 3x = \frac{20\sqrt{5} - 9\sqrt{5}}{27} = \frac{11\sqrt{5}}{27} \]

### Final Result
Thus, the value of \( \cos 3x \) is:
\[ \cos 3x = \frac{11\sqrt{5}}{27} \]