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# Precalculus: Periodic Functions

### Overview

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This course helps to build the foundational material to use mathematics as a tool to model, understand, and interpret the world around us. This is done through studying functions, their properties, and applications to data analysis. Concepts of precalculus provide the set of tools for the beginning student to begin their scientific career, preparing them for future science and calculus courses. This course is designed for all students, not just those interested in further mathematics courses. Students interested in the natural sciences, computer sciences, psychology, sociology, or similar will genuinely benefit from this introductory course, applying the skills learned to their discipline to analyze and interpret their subject material. Students will be presented with not only new ideas, but also new applications of an old subject. Real-life data, exercise sets, and regular assessments help to motivate and reinforce the content in this course, leading to learning and mastery.

### Syllabus

• Module 1: Periodic Functions
• In this course, we expand our collection of functions which we can use to model by studying periodic functions. Periodic functions are functions whose graphs repeat themselves after a certain point. It is natural to study periodic functions as many natural phenomena are repetitive or cyclical: the motion of the planets in our solar system, days of the week, seasons, and the natural rhythm of the heart. Thus, the functions introduced in this course add considerably to our ability to model physical processes. In this module, we begin by learning methods of measuring angles.
• Module 2: Right Triangle Trigonometry
• Many common phenomena have oscillatory or periodic behavior. To model this behavior requires an understanding of functions that exhibit periodic behavior like sine, cosine, and tangent. These functions are introduced using right triangles in this module, which then lets us explore their algebraic relations.
• Module 3: Sine and Cosine as Periodic Functions
• Sine and cosine are now introduced using the unit circle, which is the circle centered at the origin with radius one. This definition of our key periodic functions extends the definition originally introduced with right triangles.
• Module 4: The Tangent and Other Periodic Functions
• The most basic periodic functions, sine and cosine, were defined for all real numbers. We now study their quotients and reciprocals. However, care must be taken to ensure we do not divide by zero. In this module, we will complete our catalog of periodic functions
• Module 5: (Some) Identities of Periodic Functions
• In an effort to simplify the work involving our periodic functions, we introduce common identities. This dramatically increases their usefulness in applications. This module will emphasize the development of a small core of identities that are continually needed and can be used to determine a much larger collection. While the number of identities is small in this module, an understanding of these and how to derive others from them is essential for success as you continue your studies.

### Taught by

Joseph W. Cutrone, PhD

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