This is my first comment on your site & wanted to express my gratitude. I just hate how all my math teachers except for one(alg2 teacher) teach math in time-consuming, unnecessary, & confusing manner. I like your use of “thought experiments” & explaining the underlying concepts. It makes Math extremely simple & helps with more advanced topics that use the ones you teach as the basis. I honestly do not know of another site dedicated to teaching the underlying concepts as a means to understand the topic overall. You have been a godsend for me in math. I cant believe you aren’t way way way more popular bc of how good you are in “deciphering” the “encryption” the majority of math teachers place on math topics/concepts.
Thanks Jose, I really appreciate it!
Hi Flora, thanks for the comment and great feedback.
I probably wasn’t clear enough in the analogy. I imagine the dome (distance of 1) as a type of boundary, where sine/cosine are stuck in the dome (their max value is 1, min value -1 when facing backwards) and all other trig functions exist outside the dome and can take on nearly any value. (Technically, sec and cosec have a minimum distance of 1, so can take any value from 1 to infinity, or -1 to -infinity when facing backwards). I’d like to do a follow-up analyzing some more of the behavior, as students are often forced to graph the values of trig functions (it’s better to visualize what values they can take on).
For learning, I’d like to describe the process a bit more as well. I have a general article on my strategy (http://betterexplained.com/articles/developing-your-intuition-for-math/), for this example I started thinking about what circular objects in the real world might represent the unit circle. I thought of a dome (after too many IMAX movies maybe?) and then the screen was a natural way to represent some partial height (sine) and distance away (cosine). From there, I was able to imagine other “buildings” around me that might represent the screen in other positions – it turned out the trig functions showed up there too. A lot of it is trial and error where you hunt around for an analogy that seems to cover a few use cases (it doesn’t have to cover them all). I hope to write more about this too.
Hi Kalid,
Once again, thank you for helping everyone see how the trig functions can be applied in real life. I often feel there’s a disconnection between the contents learned in maths and my actual life, and now I can happily link them together in a more intuitive sense.
However, I do believe there’s still a fair bit of rote required despite the intuitiveness of your explanation. It’s not immediately apparent which terms equal to ONE and which ones are free to extend beyond the unit circle e.g. what I mean is that it’s not immediately obvious to associate tangent and secant to the wall example, and COtangent and COsecant to the ceiling. Perhaps you can share some insight as to how you came about the two above examples. Your epiphany or ‘aha’ moments that led you to write these examples or the thought process you went through to get to your wonderful analogy, so that as learners, we’re not overly reliant on others to come up with an effective method of learning what you’ve dubbed as the anatomy of math , as was mentioned earlier on by @Mark Ptak.
Hi Aurelio, really glad it was helpful.
Good question on the meaning of the hypotenuse. In my head, I see the hypotenuse as the “longest side of the triangle”.
When we’re in the dome, the hypotenuse [longest side] touches the dome, and is equal to the radius. When we have the secant, we’re drawing a line all the way to the wall [which extends outside the dome]. In this case, the secant is written in terms of the hypotenuse, such as 2.5 or 10. [So the secant is 250% or 1000% larger than the original hypotenuse which is stuck inside the dome. This is also the radius of the dome.]
Ok, now I got it! You’re the best Kalid, thanks for the detailed (and very quick) answer. BTW, you’ve just sold another book 
Hi Kalid,
Excellent article, thank you VERY much for sharing. My life would be so much easier in school if teachers made these relations clear.
I got confused in the “Tangent/Secant: The Wall” section, where in the diagram you say “secant = ladder “hypotenuse””, but in the embedded calculator you say “% of hyp.” where “hyp.” means the radius. So, the hypotenuse is the secant or the radius?
Your intro to this is so funny and beautiful.
You want to rewrite the science of Trigonometry?
you are confusing the trig relationship with unite circle!
you seem to be a theatre lover and only thinking of screen and your place in the theatre ! well not everyone !!
to me you seem to measure water by Grains size !!
Hi Kalid,
Longtime lurker of your site, admirer of your ability to intuit.I appreciate examples like the hellish voyage on the appendix. I found your site way after leaving college and things are finally becoming clear to me. Thanks for your passion.
Anonymous or what do you call yourself, we know you are a genius but keep it to yourself. We are satisfied with what Khalid is givining. If you so much know trig as you claimed, why have we not heard about you uptil now? Are you not aware that people don’t know mathematics? Are you telling me that that the traditional teaching style has helped maths? Now, someone comes up to help majority of us, you are not included, you are scathingly criticising. Please, go and sit down.
Good reply Theophilus. He or She is Anonymous for some reason and we got it
My guess is he or she is either one of those teachers who have been pursuing their students to memorize trig formulae OR one of those purists who make it their goal of life to keep things ‘as they were’ and so are blind to actual logic.
Hi Kalid,
I really like your articles. This is a second thank-you note from me.
I keep aside a few hours every week for exploring math, and I read this article last week during that time. Now, today, during one of those hours, I was amazed to find that the explanations were still clear in my brain. I mean, I can really visualize the sin/cos etc. Thanks a lot.
Are you a wizard?
Identifying pause and digest moments… there are several:
11 min video broken down
0 Motiv/intro
0:38 Trig as anatomy… what are the ratios
2:00 Analogy of the Dome, Screen, Distance to Screen
3:27 Think of in terms of percents
4:46 Projecting on a wall
6:17 Ramp to the ceiling
7:25 Connections: Putting it all together
8:38 “Facts” or relationships we get
10:05 Wall or Ceiling first
For those learning this, realizing utility of pausing at particular moments could be helpful.
Great video!
Usha
Wow, thank you Usha, great summary!
Kalid:
The fact that trig relations are percentages is a very good insight! I must compliment you.
I would have a modest(?) proposal: I am of an older generation, did my studies in the 50’s. We had a visual tool always at our side, an analogue calculator, a thing called a SLIDE RULE. It is based on logarithms, and you multiply on it by SLIDING one log number line scale against another. My big slide rule has trig functions and exponential functions in the form of number lines that refer to the basic log scale. It’s all out there in front, right before your eyes. I think my generation and those before us had a better visual and intuitive grasp of math because of the slide rule. Almost all calculations on a slide rule are based on proportions. They become natural if you work with a slide rule. And: you have to keep orders of magnitude straight in your head. A slide rule user must always know where they are in a calculation. You might make little mistakes in the 3rd or 4th significant digit, but never a BIG electronic calculator type of mistake.
I have never stopped using a slide rule and never really graduated to electronic calculators. I think it is worth anybody’s while who is interested in an intuitive feel for math to look into slide rules. Their are lots of them floating around and the web has lots of info on how to use them.