Precalculus: Periodic Functions

Go to Course: https://www.coursera.org/learn/precalculus-periodic-functions

Introduction

### Course Review: Precalculus: Periodic Functions on Coursera #### Overview "Precalculus: Periodic Functions" is an insightful online course offered on Coursera that lays the groundwork for understanding how mathematics can be employed to model, interpret, and navigate the complexities of the world around us. While the course is meticulously designed for students preparing for advanced studies in science and mathematics, its content is equally beneficial for anyone interested in enhancing their analytical skills and grasping essential mathematical concepts. This makes it a versatile choice for students across diverse disciplines. ### Course Structure The course is structured into **five comprehensive modules**, each building upon the knowledge established in the previous ones. Here’s a detailed look at what each module offers: #### Module 1: Periodic Functions This introductory module delves into periodic functions, fundamental in understanding the cyclical nature of various phenomena. The concept is essential not just in mathematics but also in fields such as astronomy and biology, where cyclical behaviors are prevalent. Students learn to measure angles, setting the stage for the exploration of periodic functions that play a pivotal role in modeling real-world scenarios. #### Module 2: Right Triangle Trigonometry In the second module, students discover how periodic functions are intricately linked to right triangles. Through the study of sine, cosine, and tangent, learners gain insight into how these functions model oscillatory behavior, which is foundational for understanding waves and harmonic motion. This module masterfully combines geometry with algebra, reinforcing students' mathematical competency. #### Module 3: Sine and Cosine as Periodic Functions The third module explores sine and cosine functions through the lens of the unit circle, evolving the understanding of these functions from basic definitions tied to right triangles to their broader applications. This transition is crucial for students, as grasping these functions’ behavior over the entire real number line unlocks their potential to solve complex equations and model various phenomena. #### Module 4: The Tangent and Other Periodic Functions This module expands students' toolkit by introducing other periodic functions and examining their specificity, including reciprocal and quotient forms. Emphasis on avoiding division by zero serves as a practical lesson in safeguarding mathematical operations—a fundamental skill for any mathematician or scientist. #### Module 5: Identities of Periodic Functions In the final module, students learn about common identities associated with periodic functions. The focus here is on simplifying complex relationships and fostering a deeper understanding of how to manipulate these functions in various contexts. Developing a solid grasp of these identities not only prepares students for future mathematical courses but also enhances their analytical skills in practical applications. ### Recommendations "Precalculus: Periodic Functions" is highly recommended for anyone looking to solidify their foundational knowledge in mathematics, especially in functions and trigonometry. The course is not only suitable for aspiring scientists and mathematicians but also beneficial for students in fields such as engineering, physics, and data analysis. **Pros:** - **Comprehensive Content:** The course covers all necessary concepts required for a strong precalculus foundation. - **Real-World Applications:** Emphasis on periodic functions allows students to see the relevance of math in everyday life. - **Structured Learning:** Each module builds logically on the previous one, facilitating easier comprehension. **Cons:** - **Pace:** For some learners, the course may move too quickly through advanced topics, so students are encouraged to take notes and revisit challenging sections as needed. Overall, if you're eager to demystify mathematics and intend to use it as a tool for problem-solving and modeling, enrolling in "Precalculus: Periodic Functions" on Coursera is an excellent step in your educational journey. Dive in and embrace the beauty of functions and their periodicity!

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.

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.

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.

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

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.