If sin `theta=(3)/(5),theta` lies in second quadrant, find cos `theta` and cot `theta`.

Answer

To solve the problem, we need to find the values of cos θ and cot θ given that sin θ = 3/5 and θ lies in the second quadrant.

### Step-by-Step Solution:

1. **Identify the given information**:
- We know that sin θ = 3/5.
- Since θ is in the second quadrant, sin θ is positive, and cos θ is negative.

2. **Use the Pythagorean identity**:
- The Pythagorean identity states that:
\[ \sin^2 θ + \cos^2 θ = 1 \]
- Substituting the value of sin θ:
\[ \left(\frac{3}{5}\right)^2 + \cos^2 θ = 1 \]

3. **Calculate sin² θ**:
- Calculate (3/5)²:
\[ \frac{9}{25} \]
- Now substitute this back into the identity:
\[ \frac{9}{25} + \cos^2 θ = 1 \]

4. **Solve for cos² θ**:
- Rearranging the equation gives:
\[ \cos^2 θ = 1 - \frac{9}{25} \]
- Convert 1 to a fraction with a denominator of 25:
\[ 1 = \frac{25}{25} \]
- Now subtract:
\[ \cos^2 θ = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

5. **Find cos θ**:
- Take the square root of both sides:
\[ \cos θ = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} \]
- Since θ is in the second quadrant, cos θ must be negative:
\[ \cos θ = -\frac{4}{5} \]

6. **Find cot θ**:
- Cotangent is defined as:
\[ \cot θ = \frac{\cos θ}{\sin θ} \]
- Substitute the values:
\[ \cot θ = \frac{-\frac{4}{5}}{\frac{3}{5}} = -\frac{4}{3} \]

### Final Answers:
- \( \cos θ = -\frac{4}{5} \)
- \( \cot θ = -\frac{4}{3} \)