# Conics and Trigonometry

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## Overview

Where will a cannonball land after firing? How does a harmonograph create art? What musical notes will produce a particular sine curve? Through motivating questions and interactive sliders, you’ll learn trigonometry and conics without relying on memorization.

By the end of this course, you’ll be a master in geometric shape equations, graphing an object’s motion, combining trigonometric identities, and identifying the perfect time to invest.

### Topics covered

• Circles and Ellipses
• Exponential Growth
• Hyperbolas
• Limacons
• Logarithms
• Parabolas
• Polar Coordinates
• Sine and Cosine
• Tangent and Secant
• The Unit Circle
• Trigonometric Identities
• Wave Frequency

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1. 1

### Introduction

Apply the power of experiment to learn about topics including conic sections, trigonometry, and polar coordinates.

1. #### Exploring the Sine

Start your study of trigonometry with this exploration of the sine function.

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2. #### Circles to Polar

Take the equations for circles and translate them into the language of a new type of coordinate system.

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3. #### Orbits

Stretch circles into ellipses and learn how they are applied to model planetary motion.

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2. 2

### Conic Sections

Slice a cone to get one of these!

1. #### Introduction to Conic Sections

Strengthen your intuition for conic sections and explore the parabola as a special case of conic slices.

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2. #### Circles

Practice moving circles around the coordinate plane.

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3. #### Ellipses

Stretch circles into ellipses and understand the relationship between an ellipse's foci and its function.

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4. #### Parabolas

Learn another way to define a parabolic curve using a focus point and directrix line.

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5. #### Hyperbolas

Cut the cone vertically to get this strange set of conic sections.

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3. 3

### Periodic Motion

Herein you'll play with rotations, cycles, and the beginnings of trigonometry.

1. #### Mirrors and Lenses

What happens when mirrors are shaped like parabolas?

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2. #### Projectile Motion

Study and control the paths of flying projectiles.

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3. #### Pendulums I

Learn about some of the fundamental properties of pendulums.

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4. #### Pendulums II

Let your pendulums swing in two dimensions and see how conic sections and trig explain what's going on!

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4. 4

### Trigonometry

Apply the power of the unit circle to graph trigonometry functions and combine them together.

1. #### Unit Circle Trigonometry

Learn where all of the different trigonometric functions come from.

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2. #### Frequency and Phase Shift

Explore the basic attributes of waves.

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When you add two sine functions, sometimes it makes a mess and other times it makes just another sine.

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4. #### Graphing Other Trig Functions

Practice doing calculations with the more unusual trigonometry graphs.

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5. #### Sum and Difference Formulas

Derive a formula that's fundamental to the behavior of trigonometry.

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5. 5

### Polar Graphs

Plot on a circle rather than a grid for some spectacular graphs.

1. #### Polar Intro

Explore a new method of graphing that uses angle and distance as the two independent variables.

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2. #### Basic Polar Flowers

Make flowers using sine and cosine.

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3. #### More Flowers

Build up your skills with polar graphing by creating and understanding more complex flowers.

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4. #### Cardioids and Limacons

Explore another special case — polar graphs that look like hearts and beans.

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5. #### Polar Problem Solving

Strengthen your skills by solving some challenging polar graphing problems.

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6. 6

### Exponents and Logarithms

Keep on multiplying!

1. #### Exponential Growth

See what "growing exponentially" actually means.

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2. #### Changing the Base

What happens to the equation and the graph when you change the base?

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3. #### Exponential Arithmetic

Manipulate and simplify combinations and transformations of exponential functions.

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4. #### Financial Investing

Apply exponential functions to the world of money.

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5. #### Logarithms

Take the inverse of an exponent and you'll get something called a log.

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6. #### Log Arithmetic

Learn arithmetic shortcuts and practice adding, subtracting, and multiplying logarithms.

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