How Calculus Helps Us Really Understand Trigonometry




What you'll learn
Thorough understanding of Taylor series expansion
Hands on experience with Fourier analysis
Thorough understanding of differentiation
Thorough understanding of integration
Understand the significance of fundamental theorem of Calculus
Requirements
You would need Excel to practice some of the math methods.
Description
This course will teach you Calculus in a brand new way! Traditional way to learn Calculus follows a sequential model: algebra-geometry-limit-differentiation-integration. Each piece was taught on an isolated basis. Hence it's very difficult for students to make connections. Without being able to connecting the dots, it's impossible to obtain a deep understanding about Calculus. Therefore most students walk away with random bits of memory about Calculus, they don't really have a coherent and systematic understanding of Calculus.This course will do something totally different. It starts with the ultimate use of Calculus, and approach the core of Calculus step by step in a logical sequence. We wouldn't throw you a concept out of blue, instead you will be prompted with a real problem, a problem that would encourage you to think proactively what to do next. You won't be prompted with limit tool day 1, because there was no need at that time. Instead you will face a real problem: what should we do in order to understand sine? From there, you will see gradually the need to differentiate, then integrate. You will see why the limit tool would pop up. You would gain a thorough understanding of one of Calculus' greatest tool: Taylor series expansion. You would also have hands-on experience with Fourier analysis.We believe the worst way to learn math is to follow the assembly line model where things are thrown at you without you seeing the actual need. This will prevent you from engaging with the mechanics, and learning becomes a boring course of memorizing and blind practicing. We believe the best way to learn math is to think from the end use, from the real problems, then think backwards you will understand how each piece of math snap into the right place effortlessly. And this will make your math learning experience smooth and enriched.Please join in this course if you want to have a systematic understanding of Calculus!
Overview
Section 1: Content overview
Lecture 1 What you will learn
Lecture 2 Content overview
Section 2: Introduction: An overview of Taylor series method
Lecture 3 Trigonometry: Middle ages Vs. Calculus
Lecture 4 An overview of Taylor series expansion
Lecture 5 How to get first level velocity
Lecture 6 How to get layers of velocities
Section 3: How to accumulate impacts from velocities: Integration
Lecture 7 A taste of integration
Lecture 8 How to integrate over lower level velocities
Section 4: How to get velocities: Differentiation
Lecture 9 A taste of differentiation
Lecture 10 How to differentiate a general polynomial
Lecture 11 How to differentiate sine
Lecture 12 How to differentiate cosine
Section 5: Put things together: Taylor series expansion
Lecture 13 Taylor series expansion of sine and cosine
Lecture 14 Implement Taylor series in Excel
Section 6: How sine is used in analyzing sound
Lecture 15 What is Fourier analysis
Lecture 16 How to create sound waves - Limitations of Taylor series
Lecture 17 Creating sound waves in Excel
Lecture 18 Why Fourier analysis works
Lecture 19 How to run Fourier analysis
Section 7: Resources
Lecture 20 Binomial coefficients
Lecture 21 Trig combination formula
Excel is used to implement some of the math methods discussed. While it's preferred to follow the Excel exercise to enforce the understanding, it's not required since you can just watch the video and still get a reasonable understanding.



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