An Overview of Trigonometry

There are many guides about Calculus out there, but few of them are interactive. The purpose of this guide is to introduce Calculus in a way that is interactive and enjoyable, with a large amount of resources available for the student studying Calculus. Although this guide contains a wide variety of information and practice problems, it is not intended to be a stand-alone material for an advanced Calculus course in college or at the University level, but rather a supplement.

Why Calculus?

Okay, I get it. You are probably reading this, wondering: “When am I ever going to use this stuff?” or “Just get to the Calculus already (see Chapter 2)” or “I wish I had a cheeseburger.” Calculus is important because whether you want to be an investment banker, an accountant, an engineer, a mathematician, a physicist, or even have to take a college-level course in any of those subjects, a knowledge about Calculus will get you far. Almost everything you see around you, from video games to this device you are holding, or the package you got in the mail the other day, involved using Calculus at some point in their creation. Here are some questions that Calculus can answer.

What to expect from this Guide

Although this work is written in a relatively casual tone, its purpose is to teach Calculus as efficiently as possible, while allowing the reader to truly understand the content and its applications in a challenging way, similar to a college classroom atmosphere.

The ease of understanding the ideas in this text is offset by the often challenging difficulty of its practice and example problems. The next few sections will cover what you should have mastered in Mathematics up to this point, starting from the most important concepts. Most of these are from Geometry and Trigonometry. Let’s start with the special right triangles.

The special right triangles depicted previously transition readily into an important concept you learned in Trigonometry: the unit circle. It allows you to calculate values for a few trigonometric functions in your head. Although calculators are usually a better alternative, it also gives you a base method of approximating values of nearby angles. You may have been told to memorize all the values of the unit circle, but once you memorize the two simple right triangles above, memorizing the unit circle comes naturally.

The Unit Circle

Understanding the unit circle is critical to understanding trigonometric functions such as sine, cosine, and tangent, along with their behaviors. In much of mathematics, new knowledge builds upon previous knowledge, so you must become proficient with the unit circle, and how it determines the nature of trigonometric functions.

There is going to be a pop quiz coming up soon (surprise!), so in order to prepare you for it, here are some questions you might have. Even if you don’t have them, they are good preparation for what is to come.

How do I calculate the value of inverse functions in my head?

Inverse trigonometric functions return a single angle in a chosen principal range. For example, arcsin(0) = 0 because the range of arcsin is [−π/2, π/2]. The equation sin(x) = 0 has many solutions, x = kπ for any integer k, but those are solutions to the equation, not multiple values of arcsin(0). Think of sine as the y-coordinate on the unit circle, then use the principal range to choose the inverse value.

What’s with the pi’s and the º symbols? Do I have to know both?

Degree values and radian values are both used for measuring angles. Both have different applications in the real world. Degrees are usually used for construction, architecture, and large-scale modeling, whereas radians are helpful for sciences such as physics, electrical engineering, and coastal modeling of waves. You should be familiar with how to use and interchange between both. The conversion factor is: π radians = 180º. The radians are often left out. If it is a unitless angle (i.e. there is no º symbol), assume it is in radians.

What else do I need to know?

The following tables are worth memorizing.

Note: While taking this quiz, you may want to reference the glossary definition of a unit circle to help solve many of the questions. Keep in mind which quadrant the angle is in when figuring out its value.

How do these trigonometric functions relate to each other? One of the most amazing set of formulas revealed by the unit circle is collectively known as the trigonometric identities. The first one is the most important. The other two are derived from the first trigonometric identity, which is:

${\text{sin}}^2(x)+{\text{cos}}^2(x)=1$

Dividing both sides by ${\text{sin}}^2(x)$ gives

$1+{\text{cot}}^2(x)={\text{csc}}^2(x)$

Whereas dividing the first formula by ${\text{cos}}^2(x)$ gives ${\text{tan}}^2(x)+1={\sec }^2(x)$

Keep these formulas in mind. They will be covered throughout this book very frequently.

Note: writing sin²(x) is the same as writing (sin(x))². You can see why we use the first form instead, because the second version has too many parentheses.

Angle addition formulas:

Half angle formulas:

Double Angle Formulas: