Find all solutions on the interval 0 less than or equal to 0 less than 2pi. Answer exactly.
2sin(20)=-square root 3

To solve the equation:

\[
2\sin(\theta) = -\sqrt{3}
\]

we’ll first isolate \(\sin(\theta)\) by dividing both sides by 2:

\[
\sin(\theta) = -\frac{\sqrt{3}}{2}
\]

We need to find the values of \(\theta\) for which \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) within the interval \(0 \leq \theta < 2\pi\).

### Step 1: Determine Reference Angles
The sine function equals \(-\frac{\sqrt{3}}{2}\) at angles where the reference angle is \(\frac{\pi}{3}\) (or \(60^\circ\)) in the third and fourth quadrants, where sine is negative.

### Step 2: Find Solutions
The values of \(\theta\) that satisfy \(\sin(\theta) = -\frac{\sqrt{3}}{2}\) are:

1. **Third quadrant**: \(\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}\)
2. **Fourth quadrant**: \(\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\)

### Final Answer
The solutions on the interval \(0 \leq \theta < 2\pi\) are:

\[
\theta = \frac{4\pi}{3} \text{ and } \theta = \frac{5\pi}{3}
\]

Would you like more details on any steps?

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Here are some related questions you might find useful:

1. What are the general solutions for \(\sin(\theta) = -\frac{\sqrt{3}}{2}\)?
2. How do you find reference angles for trigonometric functions?
3. What are the sine and cosine values of commonly used angles like \(\frac{\pi}{3}\)?
4. How can I solve similar equations involving cosine or tangent?
5. What strategies can help in solving trigonometric equations efficiently?

**Tip:** Remember that knowing the unit circle well can simplify solving trigonometric equations significantly.

Find all solutions on the interval 0 <= θ < 2π. Answer exactly.
2sin(θ) = -√3

Learn to solve the trigonometric equation 2sin(θ) = -√3 within the interval 0 <= θ < 2π. This solution covers key trigonometry concepts, including reference angles and the unit circle, with a step-by-step approach suitable for high school students.

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