If `sintheta=(1)/(3)`, then `costheta` 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 \( \sin^2 \theta \)

First, we need to 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 \( \cos^2 \theta \)

Rearranging the equation to solve for \( \cos^2 \theta \):

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

### Step 4: Simplify the Right Side

To simplify the right side, convert 1 into a fraction with a denominator of 9:

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

### Step 5: Take the Square Root

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

\[ \cos \theta = \sqrt{\frac{8}{9}} = \frac{\sqrt{8}}{3} = \frac{2\sqrt{2}}{3} \]

### Conclusion

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

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