AP Precalculus Practice Test: Unit 3 Question #15 When does cos(x) = 1?

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**AP Precalculus Practice Test: Unit 3, Question #15 — When does \( \cos(x) = 1 \)?**

### Problem Overview:
This question asks for the values of \( x \) where \( \cos(x) = 1 \). To answer, we need to consider the behavior of the cosine function, particularly its key points on the unit circle.

### Key Concepts:
- The **cosine function**, \( \cos(x) \), is the x-coordinate of a point on the unit circle.
- The cosine of an angle gives the horizontal distance from the origin (the center of the unit circle) to a point on the unit circle for a given angle.
- The **unit circle** is a circle with a radius of 1 centered at the origin, and as we rotate around the circle, the value of \( \cos(x) \) depends on the horizontal position of the point at that angle.

### Step-by-Step Solution:
1. **Understanding the Behavior of \( \cos(x) \):**
- The cosine function starts at \( \cos(0) = 1 \), because the point on the unit circle at \( x = 0 \) lies on the positive x-axis, which is at \( (1, 0) \).
- As we move around the unit circle, the cosine function will be equal to 1 whenever the point lies on the positive x-axis.

2. **Finding All Points Where \( \cos(x) = 1 \):**
- On the unit circle, \( \cos(x) = 1 \) at \( x = 0 \) and any integer multiple of \( 2\pi \), because the unit circle is periodic with a period of \( 2\pi \).
- Therefore, the solutions to \( \cos(x) = 1 \) are:
\[
x = 2k\pi \quad \text{where} \quad k \text{ is any integer}.
\]
This means \( \cos(x) = 1 \) whenever \( x \) is a multiple of \( 2\pi \), such as \( 0, 2\pi, 4\pi, -2\pi, \dots \).

### Final Answer:
The values of \( x \) where \( \cos(x) = 1 \) are given by:
\[
x = 2k\pi, \quad \text{where} \quad k \text{ is any integer}.
\]
This includes points like \( x = 0, 2\pi, 4\pi, -2\pi \), and so on.

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

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