AP Precalculus Practice Test: Unit 3 Question #9 Finding sin(-3pi/4) Using the Unit Circle

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 #9**
**Finding \(\sin\left(-\frac{3\pi}{4}\right)\) Using the Unit Circle**

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### **Key Concepts and Vocabulary**

1. **Unit Circle**: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine of an angle is the \(y\)-coordinate of the point on the unit circle corresponding to that angle.

2. **Sine Definition**: The sine of an angle \(\theta\) in the unit circle is the vertical coordinate of the corresponding point, or:
\[
\sin(\theta) = y
\]
where \(y\) is the \(y\)-coordinate of the point on the unit circle at that angle.

3. **Negative Angle**: When the angle is negative, the direction of rotation is clockwise, as opposed to the counterclockwise direction for positive angles.

4. **Reference Angle**: The reference angle for any given angle is the acute angle formed between the terminal side of the given angle and the nearest x-axis.

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### **Question Setup**
We are tasked with finding \(\sin\left(-\frac{3\pi}{4}\right)\) using the unit circle.

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### **Step-by-Step Solution**

1. **Determine the Direction of Rotation**:
The angle \(-\frac{3\pi}{4}\) is negative, meaning we rotate clockwise on the unit circle. To locate this angle, we start at the positive \(x\)-axis and rotate \(\frac{3\pi}{4}\) radians clockwise.

2. **Identify the Quadrant**:
A rotation of \(-\frac{3\pi}{4}\) lands us in the third quadrant. In the third quadrant, both sine and cosine values are negative.

3. **Find the Reference Angle**:
The reference angle is the acute angle between the terminal side of the given angle and the \(x\)-axis. For \(-\frac{3\pi}{4}\), the reference angle is:
\[
\text{Reference Angle} = \pi - \left|\ -\frac{3\pi}{4} \right| = \pi - \frac{3\pi}{4} = \frac{\pi}{4}
\]

4. **Use the Known Sine Value for the Reference Angle**:
From the unit circle, we know that:
\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\]
Since the sine value is negative in the third quadrant, we take the negative of this value.

5. **Final Answer**:
Therefore,
\[
\sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}
\]

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### **Final Answer**

\[
\boxed{-\frac{\sqrt{2}}{2}}
\]

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### **Purpose of the Question**
This problem assesses your ability to:
- Use the unit circle to find the sine of an angle, especially when the angle is negative.
- Recognize the reference angle and apply the correct sign for sine in different quadrants.
- Recall key values for standard angles on the unit circle, such as \(\frac{\pi}{4}\), and use them to find sine.

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

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