Determine if the given θ is a solution to the equation: 2sin(θ) − 1 = 0 . sin(θ) = 1 2 (a) Give a general formula for all the solutions. (b) List six solutions.

To determine if the given \( \theta \) is a solution to the equation \( 2\sin(\theta) - 1 = 0 \), let's proceed through the following steps:

### Step 1: Solve for \( \sin(\theta) \)
Given:
\[
2\sin(\theta) - 1 = 0
\]
we isolate \( \sin(\theta) \) by adding \( 1 \) to both sides and then dividing by \( 2 \):
\[
\sin(\theta) = \frac{1}{2}
\]

### Step 2: Determine Values of \( \theta \) that Satisfy \( \sin(\theta) = \frac{1}{2} \)
The equation \( \sin(\theta) = \frac{1}{2} \) has solutions where \( \theta = \frac{\pi}{6} \) and \( \theta = \frac{5\pi}{6} \) within the range \( [0, 2\pi) \), because sine is positive in the first and second quadrants.

### Step 3: Find the General Formula for All Solutions
The sine function is periodic with period \( 2\pi \), so the general solutions for \( \theta \) are:
\[
\theta = \frac{\pi}{6} + 2n\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2n\pi
\]
where \( n \) is any integer. This formula represents all possible solutions.

### Step 4: List Six Specific Solutions
To list six specific solutions, we can substitute different integer values for \( n \):

1. \( \theta = \frac{\pi}{6} + 2(0)\pi = \frac{\pi}{6} \)
2. \( \theta = \frac{5\pi}{6} + 2(0)\pi = \frac{5\pi}{6} \)
3. \( \theta = \frac{\pi}{6} + 2(1)\pi = \frac{13\pi}{6} \)
4. \( \theta = \frac{5\pi}{6} + 2(1)\pi = \frac{17\pi}{6} \)
5. \( \theta = \frac{\pi}{6} + 2(-1)\pi = -\frac{11\pi}{6} \)
6. \( \theta = \frac{5\pi}{6} + 2(-1)\pi = -\frac{7\pi}{6} \)

These are six specific solutions to the equation \( 2\sin(\theta) - 1 = 0 \).

---

Would you like further details or explanations on any of these steps?

### Relative Questions
1. How do you derive the general solution for a trigonometric equation involving \( \sin(\theta) \) or \( \cos(\theta) \)?
2. What role does the period of a trigonometric function play in finding all solutions?
3. How can you verify if a particular \( \theta \) is indeed a solution to a given equation?
4. Are there alternative methods to solving \( 2\sin(\theta) - 1 = 0 \)?
5. How would the solution change if the equation were \( 2\sin(\theta) + 1 = 0 \)?

### Tip
When solving trigonometric equations, always consider the periodic nature of the functions to ensure you capture all possible solutions.

Determine if the given θ is a solution to the equation: 2sin(θ) − 1 = 0. sin(θ) = 1/2 (a) Give a general formula for all the solutions. (b) List six solutions.

Understand how to determine solutions for the trigonometric equation 2sin(θ) - 1 = 0. This guide provides a general formula for all solutions and lists six specific values of θ that satisfy sin(θ) = 1/2. Suitable for high school students, this solution covers key concepts in trigonometry.

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