How To Learn Trigonometry Intuitively

This is a companion discussion topic for the original entry at

Thanks Alan, glad the analogies helped. The anatomy part helps me realize the role of trig (way to explore an alien shape) but everyone has a different takeaway :).

Kalid if u remember I messaged you regarding trigzz !! … and now seeing it … huge whoo moment you re better than my science teachers better say crammer robozz :00 !! U re genius indeed

Thanks Ansh, glad it helped! =)

In your example you need to specify which angle you want the sine of because at the moment it is ambiguous.

Whoops, thanks for the suggestion! Just updated to clarify.

Hi Khalid,
My daughter is in high school. I want her to score good in SAT exam. Do you have any package or suggestion.
Appreciated very much for your response.


Kalid, you did it again! As an engineer and programmer I use those trig identities all the time but never have they been made so succinctly clear to me. Huge aha-moment with the dome-wall-ceiling analogy.
Such a pity of all the wasted time I’ve wrestled with trig in high school :slight_smile:

Please all teachers of the world use this!

@Pravin: I don’t really have many specific test prep recommendations, unfortunately. At a high level, my approach is to gain a solid intuition for the ideas + do practice exams to make sure things are clicking. If you’re having difficulties with a certain type of problem, it’s important to look for an analogy/explanation that builds deep understanding.

@Luke: Awesome, thanks! I’d used trig a lot in school, and didn’t have the identities come together until recently (argh).

My grad stat prof said…it takes a brilliant person to see a simple concept…

Your work is brilliant, thank you.

Thanks Patrick, really appreciate it – I think there always has to be a simple explanation beneath the surface complexity. (One of my favorite Einstein quotes is that unless you can explain a topic clearly, you don’t really understand it :))

hi khalid,

A good explanation indeed. An innovative and creative presentation. Can you plz do the same for hyperbolic trigonometric functions??? Plz plz plz … I look forward for it

RE: Remember, the values are percentages. If you’re pointing at a 50-degree angle, tan(50) = 1.19. Your screen is 19% larger than the hypotenuse.

Are you sure? Should it be…
Remember, the values are percentages. If you’re pointing at a 50-degree angle, tan(50) = 1.19. Your screen is 19% larger than the Wall Distance (Radius).

@rn: Hyperbolic trig functions would be a nice follow-up :). I’m hoping to explore the implications of Euler’s formula.

@Hans: Whoops, I should have clarified – the “hypotenuse” was meant to refer to the unit circle (radius = hypotenuse = 1) but this was unclear. I’ll fix up the phrasing, thanks!

Why oh why oh why oh why don’t they teach it like this in the classroom??? Thank you so much for sharing your intuitive connections to these concepts. You would think that by now, the standard curriculum would be focused around visual learning, since every human being is a visual learner, rather than teaching concepts in a fashion designed for a robot. I am showing these pages to everyone I know that has trouble with math. There’s no reason to be afraid of the subject if it’s taught like this.

Thanks Doug, really glad it helped. Math (and any subject, really) can be so much easier to learn when we look for an approach that gets things to click deep down. Often times we try to trudge through, which works in the short term, but doesn’t build lasting understanding or enthusiasm. Just as you say, the fear of learning even “difficult” subjects can be removed – it’s a blast when things to click.

As an addition to the nice article, you can wathc the (German) videos at

  • TRI01 is about trigonometry history,
  • TRI04 is an introduction to Sine,
  • TRI07 about the unit circle,
  • TRI08 for the sine wave.

I plan to do some English versions in the future. Let me know if you like it :slight_smile:

Khalid ,
Trigonometry is used to be Tricknometry but now I find it is as simple as eating banana .Great job well done.I love your creative thinking.

Wow! This is the best I’ve ever seen. I’ve never thought in terms of percentages. It’s is little bit lengthy though, but your’re a star Kalid!

For the percentage you can also use this app: (click on Options -> Prozent Modus).

It gives you something like that:

For more playing around: or sine and cosine curves:

Have fun!