\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{\mathrm{sec} } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{\mathrm{arccot} } \) \( \newcommand{\arcsec}{\mathrm{arcsec} } \) \( \newcommand{\arccsc}{\mathrm{arccsc} } \) \( \newcommand{\sech}{\mathrm{sech} } \) \( \newcommand{\csch}{\mathrm{csch} } \) \( \newcommand{\arcsinh}{\mathrm{arcsinh} } \) \( \newcommand{\arccosh}{\mathrm{arccosh} } \) \( \newcommand{\arctanh}{\mathrm{arctanh} } \) \( \newcommand{\arccoth}{\mathrm{arccoth} } \) \( \newcommand{\arcsech}{\mathrm{arcsech} } \) \( \newcommand{\arccsch}{\mathrm{arccsch} } \)

17Calculus Precalculus - Trigonometry




Rational Functions





Complex Numbers




All of trigonometry is based on right triangles with the help of the unit circle. We have found some great sites and videos that explain trig intuitively and have lots of examples.
We do not attempt to cover all of trigonometry, just what you need for calculus.

Intuitive Understanding of Trig

It is easy to get lost in the details of trig and never figure out why we are even learning it. BetterExplained.com has a great page on understanding trig intuitively. This page will give you a feel for it and some great examples on where trig is used and why we should study it. Before going on, read and study this page thoroughly.

Right Triangles

Most of the time you will be working with right triangles, which are at the heart of trigonometry. Remember from geometry that right triangles are triangles with one of the interior angles measuring 90o. This is the same as saying that one of the sides is perpendicular to another. Also remember from geometry that when you add up all the interior angles of a triangle, you get 180o. So, if you know that one of the angles is 90o, that leaves 180 - 90 = 90 for the other two angles. This is an important thing to remember when working with right triangles. First determine which angle in the 90o angle, then work with the other two whose sum will also be 90. Pretty cool, eh?

Special Right Triangles

There are several special right triangles that you will see over and over. So it is important to become very familiar with them. They are 30-60-90 and 45-45-90 triangles. The numbers refer to the interior angles. Notice that both of them have one 90o angle. So we could also call them 30-60 and 45-45 right triangles. And, if you think about it, we really only need one angle when talking about right triangles. So by saying we have a 30 degree right triangle, we have enough information about the triangle to know that we have 30-60-90 triangle.

Here are some videos that will give us some feel for these triangles.

PatrickJMT - Special Right Triangles in Geometry: 45-45-90 and 30-60-90 [13min-13secs]

video by PatrickJMT

Krista King Math - 45-45-90 Triangles [10min-53secs]

video by Krista King Math

Krista King Math - 30-60-90 Triangles [10min-6secs]

video by Krista King Math

The Unit Circle

All of trigonometry is based on the unit circle. So it is important for you to understand the following figure and commit it to memory.

The Unit Circle [ source: wikipedia ]

Here is a good video that shows where some of the information in this figure comes from.

Krista King Math - The Unit Circle [18min-15secs]

video by Krista King Math

There is a lot of information here, so here is a video that help you remember what you need to.

PatrickJMT - A Way to Remember the Entire Unit Circle for Trigonometry [7min-18secs]

video by PatrickJMT

Trig Identities

Here is a list of the trig identities you will use most in calculus.

Set 1 - Basic Identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - Squared Identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - Double-Angle Formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\cos(2t) = \cos^2(t) - \sin^2(t)\)

Set 4 - Half-Angle Formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Remembering trig identities can be difficult. However, there are techniques to help you learn and memorize them. The main way to remember anything is to use it. It also helps to see how other people remember them. Here are some videos where the instructor explains how he remembers them. You may be able to pick up some techniques and new ideas from these videos.

PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (1) [13min-53secs]

video by PatrickJMT

PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (2) [9min-43secs]

video by PatrickJMT

PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (3) [13min-24secs]

video by PatrickJMT

Okay, so these identities may be a bit overwhelming to learn and remember. Here is a fun video that shows the geometric interpretation of all 6 trig functions. He goes through them pretty fast, so you may want to take notes but this gives you another way to remember the trig identities.

3Blue1Brown - Tattoos on Math [8min-14secs]

video by 3Blue1Brown

The Pythagorean Theorem

This is a very important theorem that you will use a lot in precalculus and calculus. Here is a good video that explains it in detail.

Krista King Math - Pythagorean Theorem [3min-7secs]

video by Krista King Math

trigonometry 17calculus youtube playlist

Really UNDERSTAND Precalculus

To bookmark this page, log in to your account or set up a free account.

Calculus Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations Topics Listed Alphabetically

Precalculus Topics Listed Alphabetically

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

learning and study techniques

Get great tutoring at an affordable price with Chegg. Subscribe today and get your 1st 30 minutes Free!

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.


Copyright © 2010-2020 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

Real Time Web Analytics
We use cookies to ensure that we give you the best experience on our website. By using this site, you agree to our Website Privacy Policy.