Solve  
sin
(
θ
)
=
−
√
3
2
  on the interval  
[
0
,
2
π
]
.
 

To solve the equation \( \sin(\theta) = -\frac{\sqrt{3}}{2} \) on the interval \([0, 2\pi]\), follow these steps:

### Step 1: Identify the reference angle
The sine of an angle equals \(-\frac{\sqrt{3}}{2}\) when the reference angle \( \alpha \) is \( \frac{\pi}{3} \). This reference angle corresponds to the angle where \( \sin(\alpha) = \frac{\sqrt{3}}{2} \), but we want the negative value, so we'll look in the appropriate quadrants.

### Step 2: Determine the quadrants
Since the sine function is negative, \( \theta \) must be in the third or fourth quadrants:
- In the third quadrant: \( \theta = \pi + \frac{\pi}{3} \)
- In the fourth quadrant: \( \theta = 2\pi - \frac{\pi}{3} \)

### Step 3: Calculate the specific angles
- In the third quadrant:
  \[
  \theta = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3}
  \]
- In the fourth quadrant:
  \[
  \theta = 2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3}
  \]

### Step 4: Provide the solution set
The solutions on the interval \([0, 2\pi]\) are:
\[
\theta = \frac{4\pi}{3}, \frac{5\pi}{3}
\]

Would you like more details or have any questions?

### Related Questions:
1. What are the values of \(\sin(\theta)\) in all four quadrants?
2. How do you determine the reference angle for a given sine value?
3. Can you explain how the unit circle is used to find these solutions?
4. What is the general solution for \(\sin(\theta) = -\frac{\sqrt{3}}{2}\)?
5. How does the periodicity of sine affect the solutions?

### Tip:
Always check which quadrants correspond to positive or negative sine values to determine the correct angles.

Solve sin(θ) = -√3/2 on the interval [0, 2π].

Learn how to solve sin(θ) = -√3/2 on the interval [0, 2π] using trigonometric concepts. This step-by-step solution covers reference angles, quadrant analysis, and key formulas for understanding the sine function on the unit circle. Perfect for Grades 10-12.

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