Thanks a lot man!.. Great explanation!
@Scott: Thanks so much for getting the interactive diagrams together, I’ve put them onto the post. (My apologies for the delay on approving that comment, it was stuck in my moderation queue because of an overactive spam filtering rule.)
@Hector: Glad you enjoyed it!
“Every circle is really the unit circle, scaled up or down to a different size.” This one did it for me. 
Very well explained, thank you
[…] And this is going to be useful in the future, but let’s think back to a triangle and our unit circle when we learned trigonometry. […]
First off…I am completely in agreement with Alecks on the insight. As I spend time presently on the calculus of integration of trigonometric functions I can’t wait to hear the insight on that one. (Found it …the downside of this site… is that I am starting to wait for Kalid to provide me the insight…must work on that).
Thanks Mark – hah, don’t want anyone getting dependent! =)
I’d like to do a follow-up on the calculus properties of the trig functions, now that I have a better understanding of them myself…
Kalid, your website has added an immense amount of intuition to my understanding of mathematics. Thank you for your fresh approach to the topics you cover. After reading this I began getting into hyperbolic trig functions. Most of which, I can’t find anything that provides much intuition on the subject. I think, since you love e so much, you could provide a lot of intuition on these functions since their definitions involve 2 terms of e. Thanks for your time.
Somehow last night I went from the triangles in the circles to the wedges formed by the secant and tangent lines. With that, the area of the 30 60 90 triangle with one leg length of 1 becomes 1/(2V3) [ one over 2 root three]. That is the beauty of the insights you provide they build up our own abilities to make new connections.
Thanks Mark, that’s a cool extension. I’ve been milling about, thinking of other intuitions that can pop out (such on the Law of Sines), hope to share them down the road too.
Thanks Andrew, I’d like to do a follow-up on hyperbolic trig functions – their connection to e is pretty neat. Also, we can even define the regular trig functions in terms of e as well :).
Marvelous!
Being a Physics teacher I have to give an insight into basic mathematics(trigs, calculus, probability, complex nos etc) to my students and I so far for trigs I use the circle/tangent analogy. But your dome analogy is far more efficient and natural.
Thanks for this wonderful insight!
Thanks Harish, really glad it helped :). [I love being able to provide teachers with new analogies to try out.]
It was only recently (i.e. a few months ago) that trig started clicking this way, I wish I’d had it as a student too!
Hey K–
This is beautiful, but how about adding, while you’re at it, the derivatives of the sin and cos functions. The explanations I’ve seen are understandable but more elaborate than intuitive.
Thanks.
Hey Tim! Great idea, I’d like to cover the derivatives of sine/cosine in a follow-up. I’m still working on a solid intuition beyond the definitions/calculus reasoning :).
Thanks Kalid,
Your triple triangle diagram and the ‘tanned gent you can see’ tip certainly clarifies the relationships of the ratios - when interpreted correctly. Also the fact that the unknown sides are percentages of the known sides is seriously illuminating.
You could shorten the explanation by cutting some of the anatomy content as well as the higher Trig references. But well done for this explanation which I am unlikely to forget anytime soon.
Alan.
a
I just love how you explained that SUC-A-TOE(A) does not work(as I call it). I have been battling this issue with teachers for years now. And yes, I do know that it’s SOH-CAH-TOA
I prefer to teach “CONCEPTS” then to just give “quick” ways to “memorize” a formula. No learning occurs!
Thanks Diana! I agree, memorizing acronyms is a poor substitute for internalizing the actual concept (they should serve as reminders, not lessons).