If x lies in III quadrant and `tan x =5/(12)`, then sin x and cos x respectively are

Options

A. `5/(13) , 12/(13)`
B. `-5/(13) , 12/(13)`
C. `5/(13), -12/(13)`
D. `-5/(13) , -12/(13)`

Answer

To solve the problem, we need to find the values of sin x and cos x given that \( x \) lies in the third quadrant and \( \tan x = \frac{5}{12} \).

### Step-by-Step Solution:

1. **Identify the Quadrant and Sign of Trigonometric Functions:**
Since \( x \) is in the third quadrant, we know that:
- \( \tan x \) is positive.
- \( \sin x \) is negative.
- \( \cos x \) is negative.

2. **Use the Definition of Tangent:**
We have \( \tan x = \frac{5}{12} \). By definition, \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \). Here, we can consider:
- Opposite side = 5
- Adjacent side = 12

3. **Calculate the Hypotenuse Using Pythagorean Theorem:**
We can find the hypotenuse \( r \) using the Pythagorean theorem:
\[ r = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

4. **Find Cosine and Sine:**
Now we can find \( \cos x \) and \( \sin x \):
\[ \cos x = \frac{\text{adjacent}}{r} = \frac{12}{13} \]
\[ \sin x = \frac{\text{opposite}}{r} = \frac{5}{13} \]

5. **Apply the Signs for the Third Quadrant:**
Since both sine and cosine are negative in the third quadrant:
\[ \cos x = -\frac{12}{13} \]
\[ \sin x = -\frac{5}{13} \]

### Final Answer:
Thus, the values of \( \sin x \) and \( \cos x \) are:
\[ \sin x = -\frac{5}{13}, \quad \cos x = -\frac{12}{13} \]