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**Question:**

"Find the general solution for each of the following trigonometric equations over the interval \( 0 \leq x < 2\pi \). For each equation:  
1) Draw a unit circle and mark the key reference angle solutions to x.  
2) Write the general solution for x.  
3) Write a separate equation for x that describes the principal solutions in the interval \( 0 \leq x < 2\pi \)."

(a) \( \sin(x) = -\frac{\sqrt{3}}{2} \)  
(b) \( \cos(x) = \frac{1}{2} \)  
(c) \( \tan(x) = -1 \)

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Let me solve each part step-by-step for you.

### (a) \( \sin(x) = -\frac{\sqrt{3}}{2} \)

1. **Unit Circle Analysis:**
   - The sine value of \(-\frac{\sqrt{3}}{2}\) corresponds to a reference angle of \( \frac{\pi}{3} \) in radians.
   - Since sine is negative in the 3rd and 4th quadrants, the angles corresponding to \(\sin(x) = -\frac{\sqrt{3}}{2}\) are:
     - \( x = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \)
     - \( x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \)

2. **General Solution for x:**
   \[
   x = \frac{4\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{3} + 2n\pi, \quad \text{where } n \in \mathbb{Z}
   \]

3. **Principal Solutions in the Interval \( 0 \leq x < 2\pi \):**
   \[
   x = \frac{4\pi}{3}, \, \frac{5\pi}{3}
   \]

### (b) \( \cos(x) = \frac{1}{2} \)

1. **Unit Circle Analysis:**
   - The cosine value of \(\frac{1}{2}\) corresponds to a reference angle of \( \frac{\pi}{3} \).
   - Since cosine is positive in the 1st and 4th quadrants, the angles are:
     - \( x = \frac{\pi}{3} \)
     - \( x = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \)

2. **General Solution for x:**
   \[
   x = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad x = \frac{5\pi}{3} + 2n\pi, \quad \text{where } n \in \mathbb{Z}
   \]

3. **Principal Solutions in the Interval \( 0 \leq x < 2\pi \):**
   \[
   x = \frac{\pi}{3}, \, \frac{5\pi}{3}
   \]

### (c) \( \tan(x) = -1 \)

1. **Unit Circle Analysis:**
   - The tangent value of \(-1\) corresponds to a reference angle of \( \frac{\pi}{4} \).
   - Since tangent is negative in the 2nd and 4th quadrants, the angles are:
     - \( x = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)
     - \( x = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \)

2. **General Solution for x:**
   \[
   x = \frac{3\pi}{4} + n\pi \quad \text{or} \quad x = \frac{7\pi}{4} + n\pi, \quad \text{where } n \in \mathbb{Z}
   \]

3. **Principal Solutions in the Interval \( 0 \leq x < 2\pi \):**
   \[
   x = \frac{3\pi}{4}, \, \frac{7\pi}{4}
   \]

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Would you like more details on any part of these solutions, or do you have any further questions? 

### Here are 5 related questions to consider:
1. What are the general solutions for trigonometric equations over all real numbers?
2. How can the unit circle be used to determine the sine, cosine, and tangent values for any angle?
3. What are the properties of the sine, cosine, and tangent functions that affect their solutions?
4. How do the signs of trigonometric functions change in different quadrants?
5. What are the key reference angles to remember for common trigonometric values?

**Tip:** For trigonometric equations, always check which quadrants your solutions fall into based on the signs of the trigonometric function.

Find the general solution for each of the following trigonometric equations over the interval 0 ≤ x < 2π. For each equation:
1) Draw a unit circle and mark the key reference angle solutions to x.
2) Write the general solution for x.
3) Write a separate equation for x that describes the principal solutions in the interval 0 ≤ x < 2π.
(a) sin(x) = -√3/2
(b) cos(x) = 1/2
(c) tan(x) = -1

Solve trigonometric equations involving sin(x) = -√3/2, cos(x) = 1/2, and tan(x) = -1 using the unit circle method. This step-by-step guide provides solutions for the general and principal values within the interval 0 ≤ x < 2π, suitable for high school students in Grades 9-12.

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