If `sin theta = 3//5,` find the values of cos`theta` and tan`theta.`

Options

A. `4/5 ; 3/4`
B. `4/5 ; 4/3`
C. `5/4 ; 3/4`
D. `4/5 ; 4/4`

Answer

To solve the problem where we are given that \( \sin \theta = \frac{3}{5} \) and we need to find \( \cos \theta \) and \( \tan \theta \), we can follow these steps:

### Step 1: Use the Pythagorean Identity
We know from trigonometric identities that:

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

### Step 2: Substitute the value of \( \sin \theta \)
Substituting \( \sin \theta = \frac{3}{5} \) into the identity:

\[ \left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \]

Calculating \( \left(\frac{3}{5}\right)^2 \):

\[ \frac{9}{25} + \cos^2 \theta = 1 \]

### Step 3: Solve for \( \cos^2 \theta \)
Rearranging the equation to solve for \( \cos^2 \theta \):

\[ \cos^2 \theta = 1 - \frac{9}{25} \]

To subtract \( \frac{9}{25} \) from 1, we convert 1 into a fraction with a denominator of 25:

\[ 1 = \frac{25}{25} \]

Now we can perform the subtraction:

\[ \cos^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

### Step 4: Find \( \cos \theta \)
Taking the square root of both sides gives us:

\[ \cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]

### Step 5: Determine the correct sign for \( \cos \theta \)
Since \( \sin \theta = \frac{3}{5} \) is positive, \( \theta \) could be in the first quadrant (where both sine and cosine are positive) or the second quadrant (where sine is positive and cosine is negative).

For the first quadrant, we take:

\[ \cos \theta = \frac{4}{5} \]

### Step 6: Find \( \tan \theta \)
Now, we can find \( \tan \theta \) using the definition of tangent:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

Substituting the known values:

\[ \tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Summary of Results
Thus, the values are:

\[ \cos \theta = \frac{4}{5} \quad \text{and} \quad \tan \theta = \frac{3}{4} \]