Trigonometry: Quickly and Intuitively

Published on Sunday, March 02, 2014 in , , , , ,

Leonardo Da Vinci's Vitruvian ManTrigonometry is a subject that can strike fear into the heart of almost any high school student.

It's actually quite understandable and useful if taught clearly enough so you grasp it. In this post, I'll show you just where to find those resources.

BASICS: As with any topics, you'll want to make sure you have a few basics down. When getting started with trigonometry, that means being clear on the concepts of similarity, and the good old Pythagorean Theorem.

The series Project Mathematics! is of great help here. Their amazing computer-animated lessons make the concepts of similarity clear and even simple:

Part 1:

Part 2:

Next, make sure you're up to speed on the good old Pythagorean Theorem, and then you'll be ready to proceed to the trigonometry of the unit circle:

THE UNIT CIRCLE: I've posted my own tutorials on the unit circle and trigonometric functions, but I truly have to tip my hat to a recently posted tutorial that far exceeds both of those.

Kalid at BetterExplained.com just posted an awesome article titled, How To Learn Trigonometry Intuitively, including a video to help you along.

What makes this post so great? He slowly introduces each concept and makes each step concrete and understandable. Kalid starts by likening trigonometry to learning anatomy, as if you were learning the anatomy of a circle. After his introduction to sine and cosine with a dome analogy, he points out that the numbers you're seeing are percentages. How much bigger or smaller is this part or that part compared to the radius of the circle? Cleverly, he even goes back to his anatomy metaphor to make this more understandable.

With the idea of a wall next to the dome, he then introduces the tangent and the secant, and then uses a ceiling built over the dome to help drive home the ideas of the cotangent and the cosecant. Probably the most startling moment in the whole post, however, is when Kalid gets you to see the connections between the 3 types of triangles he's been explaining. When you see that they're simply scaled versions of each other, everything begins to fall in place!

Once you have the scaled triangles in your mind, your knowledge of the Pythagorean Theorem and similar triangles make the relationships almost trivial to work out in your head! Instead of memorizing formulas you'll quickly forget after a trig test, you simply grasp the relationships, and can work them out anytime you need them!

Even as good as Kalid's explanations are, he points out that you shouldn't get too attached to the static diagrams. Taking that advice, I used the online graphing calculator as Desmos.com to create some models I could play around with to grasp the concepts for myself, and I've linked to them using the corresponding section names from Kalid's article:

Sine/Cosine: The Dome

Tangent/Secant: The Wall

Cotangent/Cosecant: The Ceiling

Visualize The Connections

FURTHER READING: If you enjoyed learning about trigonometry this way, there are a few other of Kalid's post I highly recommend.

First, read Surprising Uses of the Pythagorean Theorem to help you get out of the mindset that the Pythagorean Theorem is only about triangles. Next, check out How To Measure Any Distance With The Pythagorean Theorem and learn how you can use it to bring problems with a mind-boggling array of factors down to a size you can manage.

Finally, since radians are so important to the unit circle, but come across as more confusing than they should be, Kalid's Intuitive Guide to Angles, Degrees and Radians is a definite must-read.

I sincerely hope you take the time to explore most, if not all, of these resources, as they gave me a new respect and understanding for the tools of trigonometry, and I simply want to share that joy of discovery.

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1 Response to Trigonometry: Quickly and Intuitively

7:37 AM

Thanks for the PATHAGORAS THEORAM it will clear my all problems that i was facing in my trigonometry problems thanks again and hope for many more from your side.