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Part 1
Solve the equation. Give a general formula for all the solutions. List six solutions.
sine theta equals one half


The equation you are solving is:

\[
\sin(\theta) = \frac{1}{2}
\]

### Step 1: Find the General Solutions
The sine function equals \(\frac{1}{2}\) at specific angles. From the unit circle, we know that:

\[
\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\]
and
\[
\sin\left(5\frac{\pi}{6}\right) = \frac{1}{2}
\]

Because sine is a periodic function, these solutions repeat every \(2\pi\) radians. The general solutions are given by:

\[
\theta = \frac{\pi}{6} + 2n\pi \quad \text{or} \quad \theta = 5\frac{\pi}{6} + 2n\pi
\]
where \(n\) is any integer.

### Step 2: List Six Specific Solutions
Now, let's list six specific solutions by substituting integers for \(n\).

For \(\theta = \frac{\pi}{6} + 2n\pi\):
- When \(n = 0\), \(\theta = \frac{\pi}{6}\)
- When \(n = 1\), \(\theta = \frac{\pi}{6} + 2\pi = \frac{13\pi}{6}\)
- When \(n = -1\), \(\theta = \frac{\pi}{6} - 2\pi = -\frac{11\pi}{6}\)

For \(\theta = 5\frac{\pi}{6} + 2n\pi\):
- When \(n = 0\), \(\theta = 5\frac{\pi}{6}\)
- When \(n = 1\), \(\theta = 5\frac{\pi}{6} + 2\pi = \frac{17\pi}{6}\)
- When \(n = -1\), \(\theta = 5\frac{\pi}{6} - 2\pi = -\frac{7\frac{\pi}{6}}{6}\)

### Summary of Six Solutions:
1. \(\theta = \frac{\pi}{6}\)
2. \(\theta = 5\frac{\pi}{6}\)
3. \(\theta = \frac{13\pi}{6}\)
4. \(\theta = \frac{17\pi}{6}\)
5. \(\theta = -\frac{11\pi}{6}\)
6. \(\theta = -\frac{7\pi}{6}\)

These are six particular solutions to the equation.

Would you like further explanation or details on these solutions?

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### Follow-up Questions:
1. How does the unit circle help in solving trigonometric equations?
2. Why is sine periodic, and what does the period tell us?
3. Can you solve \(\sin(\theta) = -\frac{1}{2}\) using similar methods?
4. What is the relationship between radians and degrees for these angles?
5. How would you find solutions to \(\cos(\theta) = \frac{1}{2}\)?

#### Tip:
To convert between radians and degrees, recall that \( \pi \) radians equals 180 degrees!

Solve the equation. Give a general formula for all the solutions. List six solutions.
Sine theta equals one half.

Learn how to solve the trigonometric equation sin(θ) = 1/2. This step-by-step solution covers the general formula for all solutions using the unit circle and lists six specific solutions. Key concepts include trigonometry and the periodic nature of sine functions. Suitable for students in Grades 9-12.

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