AP Precalculus Practice Test: Unit 3 Question #31 Solve the Sine Function 2sin(x) = 1

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

**AP Precalculus Practice Test: Unit 3, Question #31** asks you to solve the sine equation \( 2\sin(x) = 1 \) and find the values of \(x\) where this equation is true.

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### **Key Concepts**
1. **Isolating the Sine Function**:
- Divide both sides of the equation by 2 to isolate \(\sin(x)\):
\[
\sin(x) = \frac{1}{2}
\]

2. **Sine Values on the Unit Circle**:
- The sine of an angle represents the y-coordinate of the corresponding point on the unit circle.
- The value \(\frac{1}{2}\) occurs at specific reference angles in the first and second quadrants, where sine is positive.

3. **Reference Angle**:
- The reference angle for \(\sin(x) = \frac{1}{2}\) is \(\pi/6\) or \(30^\circ\).

4. **Quadrants for Positive Sine**:
- Sine is positive in the first and second quadrants.

5. **General Solutions for Sine**:
- Solutions for \(\sin(x) = c\) can be written as:
\[
x = \text{reference angle} + 2n\pi \quad \text{(First Quadrant)}
\]
\[
x = \pi - \text{reference angle} + 2n\pi \quad \text{(Second Quadrant)}
\]
where \(n\) is any integer.

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### **Solving the Problem**
1. **Reference Angle**:
- The reference angle is \(\pi/6\).

2. **First Quadrant Solution**:
- In the first quadrant, the angle is \(x = \pi/6\).

3. **Second Quadrant Solution**:
- In the second quadrant, the angle is \(x = \pi - \pi/6 = 5\pi/6\).

4. **General Solutions**:
- Include all possible values by adding \(2n\pi\) to account for the periodic nature of sine.
- The solutions are:
\[
x = \pi/6 + 2n\pi \quad \text{and} \quad x = 5\pi/6 + 2n\pi
\]
where \(n\) is any integer.

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### **Description**:
"The equation \(2\sin(x) = 1\) simplifies to \(\sin(x) = \frac{1}{2}\). The solutions are the angles where sine equals \(\frac{1}{2}\), occurring in the first and second quadrants. The specific solutions are \(x = \pi/6\) and \(x = 5\pi/6\), with general solutions given by \(x = \pi/6 + 2n\pi\) and \(x = 5\pi/6 + 2n\pi\), where \(n\) is any integer."

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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