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The World of Trigonometry

You've probably heard of sine, cosine, and tangent before. But do you actually know what they are? While they sound like complicated ideas, they're really not. Keep on reading to learn more about the origins and uses of trigonometry.

By
Jason Marshall, PhD,
January 11, 2014
Episode #181

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The World of Right Triangles

While all triangles are beautiful, fascinating, and useful in some capacity or another, the class of very special triangles known as "right triangles" are of particular importance in the trigonometric world. As we learned when we talked about the Pythagorean Theorem, a right triangle is just a triangle in which two of its three sides come together to form a right angle.

Okay, that's great to know, but to understand what it means, we need to understand what a right angle is. Fortunately that's pretty easy. In fact, the definition is right there in the name—just look at the similarity between the term “right angle” and the word “rectangle” (rect-angle). They practically look and sound the same, right? Well, that's not by chance. After all, the four corners of a rectangle each form a right angle!

So the world of trigonometry that we're going to be talking about has to do with these special triangles known as right triangles with one right-angled corner. 

Opposite, Adjacent, and Hypotenuse

Before we go on, we quickly need to get one more thing straight—and that's our convention for how we talk about the three sides of a triangle. The idea here is pretty simple, but if you don't bother to stop and think about it for a quick second, you might never realize that it's as simple as it is—so make sure you understand this!

To make this clear, it's easiest if you draw yourself a few different right triangles to practice with (you can do this either mentally or on a piece of paper). Each right triangle has one side that's the longest. This longest side is always known as the triangle's hypotenuse—go ahead and write the word "hypotenuse" next to the appropriate side of each of your right triangles.

Now, for each triangle, draw a little arc across either of its two non-right-angled angles (it doesn't matter which). These marks indicate the particular corner of the right triangle that you're going to be working relative to. In other words, imagine you shrink yourself down and stand in this corner of the triangle—everything we're about to talk about is going to be relative to your new perspective. In particular, in addition to the hypotenuse, each triangle now has a side that's "opposite" and "adjacent" to the corner in which you're standing.

As I said before, the best way to see that this is true is by drawing a few right triangles, putting yourself in a few different non-right-angled corners, and checking out the lay of the land, as it were. So if this is at all confusing, I highly encourage you to do that before moving on. Once you've seen the view and have convinced yourself that every one of your triangles has an "opposite," "adjacent," and "hypotenuse," we're ready to (finally!) talk about those buttons on your calculator.

Wrap Up

But, unfortunately, that's all the time we have for today. Which means that we're going to have to continue the story of trigonometry next time.

In the meantime, please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!

Calculator and trigonometry images courtesy of Shutterstock.

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