If `sin theta=(1)/(3)`, hten `cos theta` will be-

Options

A. `+(8)/(9)`
B. `+-(4)/(3)`
C. `+(2sqrt2)/(3)`
D. `+-(3)/(4)`

Explanation

To find the value of \( \cos \theta \) given that \( \sin \theta = \frac{1}{3} \), we can use the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

### Step 1: Calculate \( \sin^2 \theta \)

Given \( \sin \theta = \frac{1}{3} \), we first calculate \( \sin^2 \theta \):

\[ \sin^2 \theta = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \]

### Step 2: Substitute into the Pythagorean identity

Now, substitute \( \sin^2 \theta \) into the Pythagorean identity:

\[ \frac{1}{9} + \cos^2 \theta = 1 \]

### Step 3: Solve for \( \cos^2 \theta \)

To isolate \( \cos^2 \theta \), subtract \( \frac{1}{9} \) from both sides:

\[ \cos^2 \theta = 1 - \frac{1}{9} \]

To perform the subtraction, convert 1 into a fraction with a denominator of 9:

\[ 1 = \frac{9}{9} \]

Now, perform the subtraction:

\[ \cos^2 \theta = \frac{9}{9} - \frac{1}{9} = \frac{8}{9} \]

### Step 4: Take the square root

Now, take the square root of both sides to find \( \cos \theta \):

\[ \cos \theta = \pm \sqrt{\frac{8}{9}} = \pm \frac{\sqrt{8}}{3} \]

### Step 5: Simplify \( \sqrt{8} \)

We can simplify \( \sqrt{8} \):

\[ \sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \]

Thus, we have:

\[ \cos \theta = \pm \frac{2\sqrt{2}}{3} \]

### Final Answer

Therefore, the value of \( \cos \theta \) is:

\[ \cos \theta = \pm \frac{2\sqrt{2}}{3} \]