If `sin theta = 3//5,` find the values of cosine `theta`

Options

A. `pm (8)/(9)`
B. `pm (4)/(3)`
C. `pm (2 sqrt(2))/(3)`
D. `pm (3)/(4)`

Explanation

To find the values of cosine \( \theta \) given that \( \sin \theta = \frac{3}{5} \), we can use the Pythagorean identity:

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

### Step 1: Write down the Pythagorean identity
We start with the identity:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 2: Substitute the value of \( \sin \theta \)
Since we know that \( \sin \theta = \frac{3}{5} \), we can substitute this into the equation:
\[ \cos^2 \theta = 1 - \left(\frac{3}{5}\right)^2 \]

### Step 3: Calculate \( \sin^2 \theta \)
Now, calculate \( \left(\frac{3}{5}\right)^2 \):
\[ \left(\frac{3}{5}\right)^2 = \frac{9}{25} \]

### Step 4: Substitute back into the equation
Now, substitute this back into the equation for \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - \frac{9}{25} \]

### Step 5: Convert 1 to a fraction
Convert 1 into a fraction with a denominator of 25:
\[ 1 = \frac{25}{25} \]

### Step 6: Perform the subtraction
Now, perform the subtraction:
\[ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

### Step 7: Take the square root
Now, take the square root of both sides to find \( \cos \theta \):
\[ \cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]

### Final Answer
Thus, the values of \( \cos \theta \) are:
\[ \cos \theta = \frac{4}{5} \quad \text{or} \quad \cos \theta = -\frac{4}{5} \]