Given: sin(θ)=−1/2 and cot(θ)<0. Which of the following can be the angle θ?

To determine the angle \(\theta\) based on the given information:

1. **\(\sin(\theta) = -\frac{1}{2}\):**
   - This implies that the angle \(\theta\) must be in a quadrant where sine is negative.
   - Sine is negative in the **third** and **fourth** quadrants.

2. **\(\cot(\theta) < 0\):**
   - \(\cot(\theta)\) (cotangent) is the reciprocal of \(\tan(\theta)\). Since \(\tan(\theta)\) is the ratio of \(\sin(\theta)\) to \(\cos(\theta)\), cotangent will be negative where sine and cosine have opposite signs.
   - In the **third quadrant**, both sine and cosine are negative, making \(\tan(\theta)\) positive, so \(\cot(\theta)\) is positive here.
   - In the **fourth quadrant**, sine is negative, and cosine is positive, making \(\tan(\theta)\) negative, so \(\cot(\theta)\) is negative.

From this, we conclude that \(\theta\) must be in the **fourth quadrant**.

3. **Standard reference angle:**
   - \(\sin(\theta) = \frac{1}{2}\) has a reference angle of \(30^\circ\) (or \(\frac{\pi}{6}\) radians).
   - In the fourth quadrant, the corresponding angle is \(360^\circ - 30^\circ = 330^\circ\) (or \(2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\) radians).

Thus, the angle \(\theta\) can be **\(330^\circ\)** or **\(\frac{11\pi}{6}\) radians**.

Would you like further details or have any questions?

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Here are 5 related questions:
1. What are the values of \(\cos(\theta)\) and \(\tan(\theta)\) for \(\theta = 330^\circ\)?
2. How do you calculate cotangent based on sine and cosine?
3. What is the reference angle, and how is it used to find angles in different quadrants?
4. Can you determine an angle if only the tangent is given?
5. How do the signs of trigonometric functions differ across quadrants?

**Tip:** When dealing with trigonometric functions, always identify the quadrant to determine the sign of the function.

Given: sin(θ) = -1/2 and cot(θ) < 0. Which of the following can be the angle θ?

Explore how to find the angle θ where sin(θ) = -1/2 and cot(θ) < 0. This trigonometry problem covers key concepts such as quadrant identification, sine and cotangent functions, and reference angles. Suitable for students in Grades 10-12.

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