In the third quadrant, the values of `sintheta` and `costheta` are:

Options

A. positive and negative respectively
B. negative and positive respectively
C. both positive
D. both negative

Answer

To determine the values of \( \sin \theta \) and \( \cos \theta \) in the third quadrant, we can follow these steps:

### Step 1: Understand the Quadrants
The unit circle is divided into four quadrants:
- **First Quadrant (0 to \( \frac{\pi}{2} \))**: All trigonometric ratios are positive.
- **Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \))**: \( \sin \theta \) and \( \csc \theta \) are positive, while \( \cos \theta \), \( \sec \theta \), \( \tan \theta \), and \( \cot \theta \) are negative.
- **Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \))**: Both \( \sin \theta \) and \( \cos \theta \) are negative, while \( \tan \theta \) and \( \cot \theta \) are positive.
- **Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \))**: \( \cos \theta \) and \( \sec \theta \) are positive, while \( \sin \theta \) and \( \csc \theta \) are negative.

### Step 2: Identify the Signs in the Third Quadrant
In the third quadrant, we need to focus on the signs of \( \sin \theta \) and \( \cos \theta \):
- \( \sin \theta < 0 \)
- \( \cos \theta < 0 \)

### Step 3: Conclusion
Thus, in the third quadrant, both \( \sin \theta \) and \( \cos \theta \) are negative.

### Final Answer
The values of \( \sin \theta \) and \( \cos \theta \) in the third quadrant are both negative.

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