How to Learn Algebra, Trigonometry and Precalculus

We were asked several times about the order in which one should read the books and watch the videos available on Real Not Complex. So we decided to write a series of posts - study guides - recommending different books to different people, depending on their interests and previous knowledge of mathematics.

In the first study guide we will take care of solidifying fundamentals. We will not dip our toes into higher mathematics just yet, but rather we will prepare you, so that your first encounter with Calculus, Linear Algebra or other topics is an enjoyable experience.

We will assume that you are a person who never studied university / college math, but you had more or less traditional mathematics training up to high school. Maybe you hated the subject, maybe you had horrible marks, but we will assume that you know something.

Then, depending on how competent and comfortable with math you turn out to be, we will recommend you appropriate resources.

How to Approach (Re)Learning Basics

On Real Not Complex we have a whole column devoted to fundamentals. We simply call it "Basics".

New Basics column on Real Not Complex

It covers Algebra, Trigonometry and Precalculus sections. Now - depending on how good you feel you are - we will direct you to study one of those sections.

If you were relatively comfortable with mathematics in school, We would suggest starting with Precalculus section. There might be moments where you will have to go back to Algebra or Trigonometry to refresh your memory, but that's totally normal and fine, so don't get discouraged if that happens. Simply try to pinpoint what exactly you don't know and find the appropriate topics in resources from those two sections.

If you however remember that Trigonometry was a serious problem for you in school (or you don't even remember what it was), I would recommend focusing on Trigonometry section first. At this point you might think "when would I ever use Trigonometry, it seems like such an obscure subject", but believe me - you will need it for a surprising amount of topics in the future. When you will study Calculus, you will understand that it was worth it to learn it well. After you feel comfortable with Trig, go back to studying Precalculus.

And lastly, if you find yourself being unable to do the simplest problems without getting confused or every Precalculus or Trigonometry resource turns out to be just incomprehensible in the first place, that's a sign you need to double down on Algebra section, before studying those two. Algebra section has several levels of difficulty itself, but in the later paragraphs we will explain to you exactly what to study there.

Don't be afraid to really spend a lot of time in each of those sections - the more solid your basics will be, the easier everything later will turn out.

Let's now talk about specific books in that section.

It's important to note that this is a highly opinionated recommendation. Not by accident Real Not Complex has much more resources listed in those sections. If you find yourself disliking your currently studied resource for whatever reason, don't be afraid to look around for something else. By title and the table of contents you should quite easily be able to make sure that it covers the same material. In those sections you will not find books that are too advanced for you.

Learning will be much easier if you like the resource you are studying from.

Textbooks - OpenStax

All of the topics described in the previous paragraphs have corresponding books released by OpenStax. You probably can't go wrong with them.

They have the following books:

This might seem like a lot, but don't get discouraged.

First of all, College Algebra, Algebra and Trigonometry and Precalculus are basically the same book. Multiple chapters are literally copies of each other.

According to what we have said before, out of these three books, you will want to study mostly Precalculus.

OpenStax Precalculus book

If you find yourself struggling with trigonometry sections, go back to Algebra and Trigonometry which has more elaborate introductory trigonometry chapter - "The Unit Circle: Sine and Cosine Functions".

And - as we stated before - it's perfectly normal to be confused from time to time, but if you find yourself continuously confused and struggle to go through even the initial chapters, it might be a sign that you need to take a few steps back.

In that case I would try to study Prealgebra, Elementary Algebra and Intermediate Algebra in reverse order. That is, start with Intermediate Algebra, and if you find yourself struggling, then switch to Elementary Algebra and so on, until you find a level adequate for you.

If you finished high school (no matter what were your grades), you most likely will not find yourself needing to go earlier than Intermediate Algebra, because it introduces even the most fundamental material.

Textbooks - Alternatives

OpenStax books are very popular, but if for some reason you find yourself not liking them, feel free to browse around Real Not Complex to find something that suits you more.

In particular, I can recommend Precalculus book by Carl Stitz and Jeff Zeager. I used it personally and I found it quite helpful when I was struggling with my first Calculus classes.

Stitz-Zeager Precalculus book

It's important to note that this book is basically two other books - College Algebra and College Trigonometry - combined, with very accessible initial trigonometry chapters, so if you find yourself struggling with this topic, definitely take a look at "Stitz-Zeager".

Videos

I personally enjoy studying with books and that is what I would recommend you. It's important to learn decoding mathematical concepts from a piece of paper, because often the more advanced subjects and courses are available only in textbook form (or even only in "lecture note" form).

However don't be shy to supplement your studies by watching video lectures, if they are available.

Luckily Basics section is full of high quality, acclaimed material. Both Khan Academy and Professor Leonard lectures are very well regarded by both teachers and students.

Here are the lectures from Khan Academy:

And here from Professor Leonard:

Professor Leonard videos

These video lectures can be especially helpful if you find yourself confused at any point. Since both Khan Academy and Professor Leonards videos are usually very nicely titled, it should be relatively easy to find the ones related to a specific topic that interests you.

But of course there will be absolutely no harm if you watch all of those lectures - quite the contrary!

Just don't fall into the trap of only watching lectures. You might soon realize that topics seem easy and clear when you watch a lecture, but suddenly they become confusing and difficult when you are doing the math yourself. That's why it is important to, at the very least, do exercises. The more exercises you do, the easier the topic will become, even if at the beginning it seemed borderline impossible to learn.

But what about the Geometry?

I am sure you have noticed by now, that we lied a little bit to you. Or at least we haven't told you everything.

There is one more section in the "Basics" column we did not write about yet - Geometry.

The thing is, even though Geometry is a very useful subject that will teach you how to reason logically, knowing it extremely in-depth is not really necessary when learning first subjects of higher, university-level mathematics. What is covered in Geometry is often introduced on need-to-know basis in other subjects.

If you don't get confused by what is happening in Trigonometry section or in the Analytic Geometry chapters in Precalculus books, you really do not have to spend too much time studying Geometry and you can always come back, when you find yourself in need to learn some of it.

But of course, if those mentioned sections and chapters do cause you to struggle, feel free to browse the resources in Geometry.

Perhaps just watching Khan Academy series on Geometry will be enough. And if not, you can read Elementary College Geometry by Henry Africk.

That's it for now

The Basics section acts as a preparation for studying higher, university level mathematics. Learn it well and you will have no trouble in studying more advanced material.

If you follow this guide and find yourself stuck at any point, not knowing what to do next, please let me know. I will adjust the guide accordingly, so the future readers will have even an easier time studying.

Meanwhile, in the next guide we will cover how you might approach studying your first college / university subjects.

See you then!