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The problem asks you to estimate the angle \(\theta\) in radians where \(\cos \theta = -0.5\), within the interval \(0 \leq \theta < 2\pi\). Let's work through this step by step:

### 1. Find the reference angle:
When \(\cos \theta = -0.5\), this corresponds to a known angle on the unit circle. The reference angle where \(\cos \theta = 0.5\) is \(\pi/3\) radians (or 60 degrees).

### 2. Identify in which quadrants \(\cos \theta = -0.5\):
Since cosine is negative in the second and third quadrants, we will need to calculate \(\theta\) for these quadrants.

- **In the second quadrant**: The angle is \(\pi - \pi/3 = 2\pi/3\).
- **In the third quadrant**: The angle is \(\pi + \pi/3 = 4\pi/3\).

### 3. Estimate \(\theta\):
So, the values of \(\theta\) where \(\cos \theta = -0.5\) are:
- \(\theta = 2\pi/3\)
- \(\theta = 4\pi/3\)

In decimal form:
- \(2\pi/3 \approx 2.094\)
- \(4\pi/3 \approx 4.189\)

Thus, the correct answers should be around 2.1 and 4.2 radians (rounded to one decimal place).

Would you like further clarification on any part of this process?

Here are five related questions to deepen your understanding:
1. What is the cosine function’s behavior in different quadrants?
2. How do reference angles help in finding angles in different quadrants?
3. What is the unit circle definition of cosine and sine?
4. Why is cosine negative in the second and third quadrants?
5. How can we convert between radians and degrees?

**Tip:** Always visualize the unit circle when solving trigonometric problems. It helps you understand the angle's position and sign of trigonometric values like sine and cosine.

Estimate θ if cos(θ) = -0.5 and 0 ≤ θ < 2π. Use the unit circle with increments of 0.1 units marked off along the circumference in a counterclockwise direction.

Learn how to estimate the angle θ in radians where cos(θ) = -0.5 using the unit circle, with step-by-step calculations. This trigonometry problem explores cosine in the second and third quadrants and reference angles, perfect for high school students.

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