how to learn trigonometry intuitively | betterexplained

9/26/15, 12:19 AMHow To Learn Trigonometry Intuitively | BetterExplained

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How To Learn Trigonometry Intuitively

Trig mnemonics like SOH-CAH-TOA(http://mathworld.wolfram.com/SOHCAHTOA.html) focus on computations, notconcepts:

TOA explains the tangent about as well as describes a circle. Sure, ifyoure a math robot, an equation is enough. The rest of us, with organic brains half-dedicated to vision processing, seem to enjoy imagery. And TOA evokes thestunning beauty of an abstract ratio.

I think you deserve better, and heres what made trig click for me.

Visualize a dome, a wall, and a ceilingTrig functions are percentages to the three shapes

(/)

by Kalid Azad 101 comments TweetTweet 73

+ =x2 y2 r2



Motivation: Trig Is AnatomyImagine Bob The Alien visits Earth to study our species.

Without new words, humans are hard to describe: Theres a sphere at the top, whichgets scratched occasionally or Two elongated cylinders appear to providelocomotion.

After creating specific terms for anatomy, Bob might jot down typical bodyproportions (http://en.wikipedia.org/wiki/Body_proportions):

The armspan (fingertip to fingertip) is approximately the heightA head is 5 eye-widths wideAdults are 8 head-heights tall



(http://en.wikipedia.org/wiki/Vitruvian_Man)

How is this helpful?

Well, when Bob finds a jacket, he can pick it up, stretch out the arms, and estimate theowners height. And head size. And eye width. One fact is linked to a variety ofconclusions.

Even better, human biology explains human thinking. Tables have legs, organizationshave heads, crime bosses have muscle. Our biology offers ready-made analogies thatappear in man-made creations.

Now the plot twist: you are Bob the alien, studying creatures in math-land!

Generic words like triangle arent overly useful. But labeling sine, cosine, andhypotenuse helps us notice deeper connections. And scholars might study haversine,exsecant and gamsin (http://www.theonion.com/articles/nations-math-teachers-



introduce-27-new-trig-functi,33804/), like biologists who find a link between yourfibia and clavicle.

And because triangles show up in circles

and circles appear in cycles, our triangle terminology helps describe repeatingpatterns!

Trig is the anatomy book for math-made objects. If we can find a metaphoricaltriangle, well get an armada of conclusions for free.

Sine/Cosine: The DomeInstead of staring at triangles by themselves, like a caveman frozen in ice, imaginethem in a scenario, hunting that mammoth.

Pretend youre in the middle of your dome, about to hang up a movie screen. Youpoint to some angle x, and thats where the screen will hang.



The angle you point at determines:

sine(x) = sin(x) = height of the screen, hanging like a signcosine(x) = cos(x) = distance to the screen along the ground [cos ~ how close]the hypotenuse, the distance to the top of the screen, is always the same

Want the biggest screen possible? Point straight up. Its at the center, on top of yourhead, but its big dagnabbit.

Want the screen the furthest away? Sure. Point straight across, 0 degrees. The screenhas 0 height at this position, and its far away, like you asked.

The height and distance move in opposite directions: bring the screen closer, and itgets taller.

Tip: Trig Values Are PercentagesNobody ever told me in my years of schooling: sine and cosine are percentages.They vary from +100% to 0 to -100%, or max positive to nothing to max negative.

Lets say I paid $14 in tax. You have no idea if thats expensive. But if I say I paid 95%in tax, you know Im getting ripped off.



An absolute height isnt helpful, but if your sine value is .95, I know youre almost atthe top of your dome. Pretty soon youll hit the max, then start coming down again.

How do we compute the percentage? Simple: divide the current value by themaximum possible (the radius of the dome, aka the hypotenuse).

Thats why were told Sine = Opposite / Hypotenuse. Its to get a percentage! Abetter wording is Sine is your height, as a percentage of the hypotenuse. (Sinebecomes negative if your angle points underground. Cosine becomes negative whenyour angle points backwards.)

Lets simplify the calculation by assuming were on the unit circle (radius 1). Now wecan skip the division by 1 and just say sine = height.

Every circle is really the unit circle, scaled up or down to a different size. So work outthe connections on the unit circle and apply the results to your particular scenario.

Try it out: plug in an angle and see what percent of the height and width it reaches:

Sine and CosineR1 x = 30 30R2 sine 50 % of heightR3 cosine 86.60254037844 % of width

The growth pattern of sine isnt an even line. The first 45 degrees cover 70% of theheight, and the final 10 degrees (from 80 to 90) only cover 2%.

This should make sense: at 0 degrees, youre moving nearly vertical, but as you get tothe top of the dome, your height changes level off.

Tangent/Secant: The Wall



One day your neighbor puts up a wall right next to your dome. Ack, your view! Yourresale value!

But can we make the best of a bad situation?

Sure. What if we hang our movie screen on the wall? You point at an angle (x) andfigure out:

tangent(x) = tan(x) = height of screen on the walldistance to screen: 1 (the screen is always the same distance along the ground,right?)secant(x) = sec(x) = the ladder distance to the screen

We have some fancy new vocab terms. Imagine seeing the Vitruvian TANGENTleman projected on the wall. You climb the ladder, making sure you can SEE,CANT you?. (Yeah, hes naked wont forget the analogy now, will you?)

Lets notice a few things about tangent, the height of the screen.



It starts at 0, and goes infinitely high. You can keep pointing higher and higheron the wall, to get an infinitely large screen! (Thatll cost ya.)

Tangent is just a bigger version of sine! Its never smaller, and while sine topsoff as the dome curves in, tangent keeps growing.

How about secant, the ladder distance?

Secant starts at 1 (ladder on the floor to the wall) and grows from thereSecant is always longer than tangent. The leaning ladder used to put up thescreen must be longer than the screen itself, right? (At enormous sizes, when theladder is nearly vertical, theyre close. But secant is always a smidge longer.)

Remember, the values are percentages. If youre pointing at a 50-degree angle, tan(50)= 1.19. Your screen is 19% larger than the distance to the wall (the radius of thedome).

Tangent and SecantR1 x = 50 50R2 tangent 119.17535925942 % of hyp.R3 secant 155.57238268604 % of hyp.

(Plug in x=0 and check your intuition that tan(0) = 0, and sec(0) = 1.)

Cotangent/Cosecant: The CeilingAmazingly enough, your neighbor now decides to build a ceiling on top of your dome,far into the horizon. (Whats with this guy? Oh, the naked-man-on-my-wall incident)

Well, time to build a ramp to the ceiling, and have a little chit chat. You pick an angleto build and work out:



cotangent(x) = cot(x) = how far the ceiling extends before we connectcosecant(x) = csc(x) = how long we walk on the rampthe vertical distance traversed is always 1

Tangent/secant describe the wall, and COtangent and COsecant describe the ceiling.

Our intuitive facts are similar:

If you pick an angle of 0, your ramp is flat (infinite) and never reachers theceiling. Bummer.The shortest ramp is when you point 90-degrees straight up. The cotangent is0 (we didnt move along the ceiling) and the cosecant is 1 (the ramp length isat the minimum).

Visualize The ConnectionsA short time ago I had zero intuitive conclusions about the cosecant. But with thedome/wall/ceiling metaphor, heres what we see:



Whoa, its the same triangle, just scaled to reach the wall and ceiling. We have verticalparts (sine, tangent), horizontal parts (cosine, cotangent), and hypotenuses (secant,cosecant). (Note: the labels show where each item goes up to. Cosecant is the fulldistance from you to the ceiling.)

Now the magic. The triangles have similar facts:



From the Pythagorean Theorem (http://betterexplained.com/articles/surprising-uses-of-the-pythagorean-theorem/) ( ) we see how the sides of each triangle arelinked.

And from similarity (http://betterexplained.com/articles/understanding-why-similarity-works/), ratios like height to width must be the same for these triangles.(Intuition: step away from a big triangle. Now it looks smaller in your field of view, butthe internal ratios couldnt have changed.)

This is how we find out sine/cosine = tangent/1.

Id always tried to memorize these facts, when they just jump out at us whenvisualized. SOH-CAH-TOA is a nice shortcut, but get a real understanding first!

Gotcha: Remember Other AnglesPsst dont over-focus on a single diagram, thinking tangent is always smaller than 1.If we increase the angle, we reach the ceiling before the wall:

+ =a2 b2 c2



The Pythagorean/similarity connections are always true, but the relative sizes canvary.

(But, you might notice that sine and cosine are always smallest, or tied, since theyretrapped inside the dome. Nice!)

Summary: What Should We Remember?For most of us, Id say this is enough:

Trig explains the anatomy of math-made objects, such as circles and repeatingcyclesThe dome/wall/ceiling analogy shows the connections between the trigfunctionsTrig functions return percentages, that we apply to our specific scenario



You dont need to memorize , except for silly tests that mistake triviafor understanding. In that case, take a minute to draw the dome/wall/ceiling diagram,fill in the labels (a tan gentleman you can see, cant you?), and create a cheatsheet(http://www.mathwords.com/t/trig_identities.htm) for yourself.

In a follow-up, well learn about graphing, complements, and using Eulers Formula(http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/) tofind even more connections.

Appendix: The Original Definition Of TangentYou may see tangent defined as the length of the tangent line from the circle to the x-axis (geometry buffs can work this out).

As expected, at the top of the circle (x=90) the tangent line can never reach the x-axisand is infinitely long.

I like this intuition because it helps us remember the name tangent, and heres anice interactive trig guide (http://www.touchmathematics.org/topics/trigonometry) toexplore:

+ =12 cot2 csc2



(http://www.touchmathematics.org/topics/trigonometry)

Still, its critical to put the tangent vertical and recognize its just sine projected onthe back wall (along with the other triangle connections).

Appendix: Inverse FunctionsTrig functions take an angle and return a percentage. means a 30-degreeangle is 50% of the max height.

The inverse trig functions let us work backwards, and are written or (arcsine), and often written asin in various programming languages.

If our height is 25% of the dome, whats our angle?

ArcsineR1 height = .25 0.25R2 asin(height) 14.47751218593+5 rowsClear

Now what about something exotic, like inverse secant? Often times its not availableas a calculator function (even the one I built, sigh).

Looking at our trig cheatsheet, we find an easy ratio where we can compare secant to1. For example, secant to 1 (hypotenuse to horizontal) is the same as 1 to cosine:

sin(30) = .5

sin1 arcsin



Suppose our secant is 3.5, i.e. 350% of the radius of the unit circle. Whats the angle tothe wall?

Appendix: A Few ExamplesExample: Find the sine of angle x.

Ack, what a boring question. Instead of find the sine think, Whats the height as apercentage of the max (the hypotenuse)?.

First, notice the triangle is backwards. Thats ok. It still has a height, in green.

Whats the max height? By the Pythagorean theorem, we know

Ok! The sine is the height as a percentage of the max, which is 3/5 or .60.



Follow-up: Find the angle.

Of course. We have a few ways. Now that we know sine = .60, we can just do:

Heres another approach. Instead of using sine, notice the triangle is up against thewall, so tangent is an option. The height is 3, the distance to the wall is 4, so thetangent height is 3/4 or 75%. We can use arctangent to turn the percentage back intoan angle:

Example: Can you make it to shore?

Youre on a boat with enough fuel to sail 2 miles. Youre currently .25 miles fromshore. Whats the largest angle you could use and still reach land? Also, the onlyreference available is Huberts Compendium of Arccosines, 3rd Ed. (Truly, a hellishvoyage.)



Ok. Here, we can visualize the beach as the wall and the ladder distance to thewall is the secant.

First, we need to normalize everything in terms of percentages. We have 2 / .25 = 8hypotenuse units worth of fuel. So, the largest secant we could allow is 8 times thedistance to the wall.

Wed like to ask What angle has a secant of 8?. But we cant, since we only have abook of arccosines.

We use our cheatsheet diagram to relate secant to cosine: Ah, I see that sec/1 =1/cos, so

A secant of 8 implies a cosine of 1/8. The angle with a cosine of 1/8 is arccos(1/8) =82.8 degrees, the largest we can afford.

Not too bad, right? Before the dome/wall/ceiling analogy, Id be drowning in a mess ofcomputations. Visualizing the scenario makes it simple, even fun, to see which trigbuddy can help us out.

In your problem, think: am I interested in the dome (sin/cos), the wall (tan/sec), or theceiling (cot/csc)?

Happy math.



Update: The owner of Grey Matters (http://gmmentalgym.blogspot.com/) puttogether interactive diagrams for the analogies (drag the slider on the left to changethe angle):

Sine/Cosine: The Dome (https://www.desmos.com/calculator/0uyr4ywrvt)Tangent/Secant: The Wall (https://www.desmos.com/calculator/2ehsvswurj)Cotangent/Cosecant: The Ceiling(https://www.desmos.com/calculator/1bswcagm9k)Combined visualization (https://www.desmos.com/calculator/az45nwnmis)

(https://www.desmos.com/calculator/az45nwnmis)

Thanks!

Other Posts In This Series1. How To Learn Trigonometry Intuitively2. Intuition For The Law Of Sines (http://betterexplained.com/articles/law-of-



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Aha!

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sines/)3. Intuition For The Law Of Cosines (http://betterexplained.com/articles/law-of-

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Posted in Guides (http://betterexplained.com/articles/category/guides/), Math(http://betterexplained.com/articles/category/math/)

Questions & Contributions

Translation Russian Translation (http://ru.yasno.tv/article/math/43-sinus-i-cosinus-eto-procenty)

Resource Tangent/cotangent touches the circle once, secant/cosecant touches ittwice. (http://curiouscheetah.com/BlogMath/basic-trig-functions/)

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101 comments1. Juliet Cooke says:

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

2. kalid says:Whoops, thanks for the suggestion! Just updated to clarify.

3. Pravin Patel says:Hi Khalid,My daughter is in high school. I want her to score good in SAT exam. Do you haveany package or suggestion.Appreciated very much for your response.Regards,

Pravin

4. Luke says:



Kalid, you did it again! As an engineer and programmer I use those trig identitiesall 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 Ive wrestled with trig in high school

Please all teachers of the world use this!

5. kalid says:@Pravin: I dont 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 youre having difficultieswith a certain type of problem, its important to look for an analogy/explanationthat builds deep understanding.

@Luke: Awesome, thanks! Id used trig a lot in school, and didnt have theidentities come together until recently (argh).

6. patrick says:My grad stat prof saidit takes a brilliant person to see a simple concept

Your work is brilliant, thank you.

7. kalid says:Thanks Patrick, really appreciate it I think there always has to be a simpleexplanation beneath the surface complexity. (One of my favorite Einstein quotes isthat unless you can explain a topic clearly, you dont really understand it :))

8. rn koushik says:hi khalid,



A good explanation indeed. An innovative and creative presentation. Can you plzdo the same for hyperbolic trigonometric functions????? Plz plz plz .. I lookforward for it

9. Hans says:RE: Remember, the values are percentages. If youre 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 youre pointing at a 50-degree angle,tan(50) = 1.19. Your screen is 19% larger than the Wall Distance (Radius).

10. kalid says:@rn: Hyperbolic trig functions would be a nice follow-up :). Im hoping to explorethe implications of Eulers formula.

@Hans: Whoops, I should have clarified the hypotenuse was meant to refer tothe unit circle (radius = hypotenuse = 1) but this was unclear. Ill fix up thephrasing, thanks!

11. Doug Bennett says:Why oh why oh why oh why dont they teach it like this in the classroom??? Thankyou so much for sharing your intuitive connections to these concepts. You wouldthink that by now, the standard curriculum would be focused around visuallearning, since every human being is a visual learner, rather than teaching conceptsin a fashion designed for a robot. I am showing these pages to everyone I know thathas trouble with math. Theres no reason to be afraid of the subject if its taughtlike this.



12. kalid says:Thanks Doug, really glad it helped. Math (and any subject, really) can be so mucheasier 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 doesntbuild lasting understanding or enthusiasm. Just as you say, the fear of learningeven difficult subjects can be removed its a blast when things to click.

13. Kai (http://www.echteinfach.tv/) says:As an addition to the nice article, you can wathc the (German) videos athttp://www.youtube.com/watch?v=yWDxBnc6XRU&list=PL63A6385F43C725CC(http://www.youtube.com/watch?v=yWDxBnc6XRU&list=PL63A6385F43C725CC)

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

14. krishnamoorthy says:Khalid ,Trigonometry is used to be Tricknometry but now I find it is as simple as eatingbanana .Great job well done.I love your creative thinking.

15. Johan Bester says:Wow! This is the best Ive ever seen. Ive never thought in terms of percentages. Itsis little bit lengthy though, but yourre a star Kalid!

16. Kai Mathematik (http://www.echteinfach.tv/) says:



For the percentage you can also use this app: http://www.echteinfach.tv/flash/?app=a0102 (http://www.echteinfach.tv/flash/?app=a0102) (click on Options ->Prozent Modus).

It gives you something like that: http://i.imgur.com/k7SUY1l.png(http://i.imgur.com/k7SUY1l.png)

For more playing around: http://www.echteinfach.tv/trigonometrie/sinus-kosinus#p (http://www.echteinfach.tv/trigonometrie/sinus-kosinus#p) or sine andcosine curves: http://www.echteinfach.tv/trigonometrie/trigonometrische-funktionen#p (http://www.echteinfach.tv/trigonometrie/trigonometrische-funktionen#p)

Have fun!Kai

17. Tom says:Im a programmer and I love algebra, but I gave up hope on ever understandingtrigonometry beyond simply memorizing percentages. I never realized it was thissimple.

Thank you.

18. Bhadrasheel says:Hi Khalid,Keep helping the world in this Wonderful way.Thank you very much.Always waiting for ur new ideas in simplifying the concept.

19. Ellie says:



Khalid,

Thanks for the lovely explanation.

Even with the wall concept, its not intuitive why Tan(x) is a positive function inthe 3rd Quadrant. That Tan(x) = Y component / X component or Sine(x)/Cos(x)kinda explains it that both quantities are negative in the 3rd quadrant and hencethe Tan function, which is a ratio, is positive. However, when you visualise the Tanfunction in the 3rd Quadrant, intuitively it feels like it should be negative.

Added complication is that if you take the word TANGENT literally as a slope of thecircle then at 90 degrees the slope of the tangent should be zero. But Tan functionis undefined at 90 degrees (division by zero at this point).

While if you visualize Tan as a magnitude/ length of the tangent, then it ought tobe negative in the 3rd Quadrant. How can I visualize this better?

20. kalid says:@Kai: Always interested in checking out resources, though Ill have to brush up(i.e. learn) some German!

@krishnamoorthy: Thanks so much. I do think most ideas can be as simple asfalling off a log if seen the right way.

@Johan: Thanks! Hah, you should have seen the original post, which was abouttwice as big :). Ill be doing a follow-up with some of that content.

@Tom: Thanks. I was in the same boat, thinking I had to memorize everything. Itsalmost like refactoring ugly code, sometimes theres a simpler way to think aboutan existing problem which makes everything snap together.

@Bhadrasheel: Appreciate it!



@Ellie: Great question how the trig angles behave in other quadrants issomething Id like to cover in the follow-up. (Article was getting big, somethinggood for the follow-up!)

Using the percentage analogy, tangent is the height relative to the wall distance,but each component can have a sign:

Is the wall in front (positive) or behind (negative)

Is the height above ground (positive) or below ground (negative)

For example,

* For x = -30, we are pointing underground so the tangent is negative.* For x = 120, we are pointing backward. The height is positive, but we are on theback wall so its negative.* For x = 210, we are pointing underground AND backward. So this is negativeheight on the back wall, which counts as positive

This matches the signs for cosine (front wall / back wall) and sine (above ground /underground) so the calculations are the same :).

21. Kai Mathematik (http://www.echteinfach.tv/) says:@kalid: Actually I am doing the same that you are doing, breaking everythingdown, not taking formula as-is, trying to find the insights behind but just inGerman and a tiny bit more animated !

Some English speakers have asked me already to transfer my videos into English. Ithink I will give it a try this year, if I find time. I will send you a message as soon asthe first video is ready. Just remember Echt Einfach TV (which means Real SimpleTV).



Kind regards!Kai

22. Scott (http://headinside.blogspot.com) says:Thank you for this wonderful intuitive explanation, Kalid! Youve done it again!

While I could follow the explanations, I did want to follow your advice and not gettoo hung up on an individual diagram. I also wanted to play around with theconcepts, so I put together the following demos on the online Desmos calculator:

Sine/Cosine: The Dome:https://www.desmos.com/calculator/0uyr4ywrvt(https://www.desmos.com/calculator/0uyr4ywrvt)

Tangent/Secant: The Wall:https://www.desmos.com/calculator/2ehsvswurj(https://www.desmos.com/calculator/2ehsvswurj)

Cotangent/Cosecant: The Ceiling:https://www.desmos.com/calculator/1bswcagm9k(https://www.desmos.com/calculator/1bswcagm9k)

Visualize The Connections:https://www.desmos.com/calculator/az45nwnmis(https://www.desmos.com/calculator/az45nwnmis)

Putting these demos together and seeing the results also helped make everythingclearer, and I thought others might find these useful.

Thanks again, Kalid!



23. joe says:hi, thanks for another great article. my only suggestion would be the large trianglein visualising connections and the others is that it (can) look as though each labelis a bit of the line and hence the total length is the sum of all the functions i takeit you mean when you go to this point, you use this function to give you thelength ie, step up from say, 1 to csc to sec NOT (1+csc+sec) = lenth of line. not abiggy but if it caught me out for a couple minutes, it might end up a road-block forsomeone else.

24. Chris (https://twitter.com/MrAdamsProblems) says:I saw this last night before going to bed, and used it this morning with myGeometry class as we began our Trig unit today.

After reading it last night, presenting it to the kids this morning, and readingthrough this again, trigonometry finally makes intuitive sense to me.

I am confident that this will help my students see this in a clearer light, andhopefully the handout that I put together to introduce sine and cosine today ishelping them make meaningful connections.

https://docs.google.com/document/d/10NdKM6hAAnOtf0yATknNcfzHWoC2Pz4A9oklIz4tmw4/edit?usp=sharing(https://docs.google.com/document/d/10NdKM6hAAnOtf0yATknNcfzHWoC2Pz4A9oklIz4tmw4/edit?usp=sharing)

25. Ellie says:That helps Khalid! Thanks a ton!

26. kalid says:@kai: Sounds great, let me know and Ill check them out.



@joe: Great feedback, Ill see if I can add a note to clarify. When youre making thediagrams you tend to have all sorts of unstated assumptions which arent there forother people :).

@Chris: Thats so awesome, I love it when the analogies come in handy forteaching. I really like how youve worked percentages into the worksheet, it puts ameaning behind the calculation (3/5 oh, thats 60%!).

@Ellie: Happy to help!

27. EJ says:This comment is to Ellie: why tangent is positive in the 3rd quadrant.

Note that tangent is NOT the slope of the circle but the slope(=rise/run=sine/cosine) of the radius extending from the center to the unit circle.When that radius is extending to the 3rd quadrant, the slope remains the same(sign and size).

Similarly, tangent is negative in the 2nd and 4th quadrants.

28. Ananya Muddukrishna says:Kalid, you have a beautiful way of explaining things. Your illustrations, intuitionbuildup and Aha! moments produce a snapping feeling in my brain. Everything justfalls into place never to be forgotten again.

I think that your articles are an invaluable gift to mankind. Keep it up! All the best.

29. kalid says:Thanks Ananya, I really appreciate the encouragement! Really glad that everythingclicked :).



30. vinay says:I am confused about the ceiling diagram. How come height traversed is always 1?Seems height can be bigger as the line extends beyond the dome.

31. Kalid says:Hi vinay, try this interactive calculator for an example:https://www.desmos.com/calculator/1bswcagm9k(https://www.desmos.com/calculator/1bswcagm9k)

When building a ramp up to the ceiling, the distance we travel depends on theangle we pick. However, the ceiling itself is always 1 unit above the ground. (In abuilding the ceiling is always a constant height, no matter how steep the stairs areto get there.)

32. Rick cruz says:Enjoyed another door I needed to open. Thanks.

33. Kalid says:Thanks Rick, glad you enjoyed it.

34. Rodrigo Alexandre Pgoli says:Dear friend, Thank you for this precious point of view about trigonometry.It was a pleasure reading your article. It was amazing!Congratulations! Thanks a lot!

35. kalid says:Thank you Rodrigo!



36. Hctor says:Thanks a lot man!.. Great explanation!

37. kalid says:@Scott: Thanks so much for getting the interactive diagrams together, Ive putthem onto the post. (My apologies for the delay on approving that comment, it wasstuck in my moderation queue because of an overactive spam filtering rule.)

@Hector: Glad you enjoyed it!

38. Aleks says:Every circle is really the unit circle, scaled up or down to a different size. This onedid it for me.

39. jodie says:Very well explained, thank you

40. kalid says:@Aleks: Awesome, glad it clicked

@jodie: Youre welcome!

41. mark ptak (http://www.we4dkids.com) says:First offI am completely in agreement with Alecks on the insight. As I spend timepresently on the calculus of integration of trigonometric functions I cant wait tohear the insight on that one. (Found it .the downside of this site is that I amstarting to wait for Kalid to provide me the insight.must work on that).



42. kalid says:Thanks Mark hah, dont want anyone getting dependent! =)

Id like to do a follow-up on the calculus properties of the trig functions, now that Ihave a better understanding of them myself

43. Andrew says:Kalid, your website has added an immense amount of intuition to myunderstanding of mathematics. Thank you for your fresh approach to the topicsyou cover. After reading this I began getting into hyperbolic trig functions. Most ofwhich, I cant 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 functionssince their definitions involve 2 terms of e. Thanks for your time.

44. mark ptak (http://www.we4dkids.com) says:Somehow last night I went from the triangles in the circles to the wedges formedby the secant and tangent lines. With that, the area of the 30 60 90 triangle withone leg length of 1 becomes 1/(2V3) [ one over 2 root three]. That is the beauty ofthe insights you provide they build up our own abilities to make new connections.

45. kalid says:Thanks Mark, thats a cool extension. Ive been milling about, thinking of otherintuitions that can pop out (such on the Law of Sines), hope to share them downthe road too.

46. kalid says:Thanks Andrew, Id like to do a follow-up on hyperbolic trig functions theirconnection to e is pretty neat. Also, we can even define the regular trig functions interms of e as well :).



47. Ansh choubey says:Kalid if u remember I messaged you regarding trigzz !! .. and now seeing it hugewhoo moment you re better than my science teachers better say crammer robozz:00 !! U re genius indeed

48. kalid says:Thanks Ansh, glad it helped! =)

49. Harish Dobhal says: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 thecircle/tangent analogy. But your dome analogy is far more efficient and natural.

Thanks for this wonderful insight!

50. kalid says:Thanks Harish, really glad it helped :). [I love being able to provide teachers withnew analogies to try out.]

It was only recently (i.e. a few months ago) that trig started clicking this way, I wishId had it as a student too!

51. Tim McGrath says:Hey K



This is beautiful, but how about adding, while youre at it, the derivatives of the sinand cos functions. The explanations Ive seen are understandable but moreelaborate than intuitive.

Thanks.

52. kalid says:Hey Tim! Great idea, Id like to cover the derivatives of sine/cosine in a follow-up.Im still working on a solid intuition beyond the definitions/calculus reasoning :).

53. Alan Williams says:Thanks Kalid,

Your triple triangle diagram and the tanned gent you can see tip certainly clarifiesthe relationships of the ratios when interpreted correctly. Also the fact that theunknown sides are percentages of the known sides is seriously illuminating.

You could shorten the explanation by cutting some of the anatomy content as wellas the higher Trig references. But well done for this explanation which I amunlikely to forget anytime soon.

Alan.

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

55. Diane says:



I just love how you explained that SUC-A-TOE(A) does not work(as I call it). I havebeen battling this issue with teachers for years now. And yes, I do know that itsSOH-CAH-TOAI prefer to teach CONCEPTS then to just give quick ways to memorize aformula. No learning occurs!

56. kalid says:Thanks Diana! I agree, memorizing acronyms is a poor substitute for internalizingthe actual concept (they should serve as reminders, not lessons).

57. jose says:This is my first comment on your site & wanted to express my gratitude. I just hatehow all my math teachers except for one(alg2 teacher) teach math in time-consuming, unnecessary, & confusing manner. I like your use of thoughtexperiments & explaining the underlying concepts. It makes Math extremelysimple & 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 underlyingconcepts as a means to understand the topic overall. You have been a godsend forme in math. I cant believe you arent way way way more popular bc of how goodyou are in deciphering the encryption the majority of math teachers place onmath topics/concepts.

58. kalid says:Thanks Jose, I really appreciate it!

59. Flora says:



Hi Kalid,Once again, thank you for helping everyone see how the trig functions can beapplied in real life. I often feel theres a disconnection between the contentslearned in maths and my actual life, and now I can happily link them together in amore intuitive sense.

However, I do believe theres still a fair bit of rote required despite the intuitivenessof your explanation. Its not immediately apparent which terms equal to ONE andwhich ones are free to extend beyond the unit circle e.g. what I mean is that its notimmediately obvious to associate tangent and secant to the wall example, andCOtangent and COsecant to the ceiling. Perhaps you can share some insight as tohow you came about the two above examples. Your epiphany or aha moments thatled you to write these examples or the thought process you went through to get toyour wonderful analogy, so that as learners, were not overly reliant on others tocome up with an effective method of learning what youve dubbed as the anatomyof math , as was mentioned earlier on by @Mark Ptak.

60. kalid says:Hi Flora, thanks for the comment and great feedback.

I probably wasnt clear enough in the analogy. I imagine the dome (distance of 1) asa 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 thedome and can take on nearly any value. (Technically, sec and cosec have aminimum distance of 1, so can take any value from 1 to infinity, or -1 to -infinitywhen facing backwards). Id like to do a follow-up analyzing some more of thebehavior, as students are often forced to graph the values of trig functions (itsbetter to visualize what values they can take on).



For learning, Id like to describe the process a bit more as well. I have a generalarticle on my strategy (http://betterexplained.com/articles/developing-your-intuition-for-math/ (http://betterexplained.com/articles/developing-your-intuition-for-math/)), for this example I started thinking about what circularobjects in the real world might represent the unit circle. I thought of a dome (aftertoo many IMAX movies maybe?) and then the screen was a natural way torepresent some partial height (sine) and distance away (cosine). From there, I wasable to imagine other buildings around me that might represent the screen inother positions it turned out the trig functions showed up there too. A lot of it istrial and error where you hunt around for an analogy that seems to cover a few usecases (it doesnt have to cover them all). I hope to write more about this too.

61. rregan (http://educationwithapurpose.com) says:Your intro to this is so funny and beautiful.

62. kalid says:@rregan: Thanks, glad you liked it :).

63. Aurelio Jargas (http://aurelio.net) says:Hi Kalid,

Excellent article, thank you VERY much for sharing. My life would be so mucheasier in school if teachers made these relations clear.

I got confused in the Tangent/Secant: The Wall section, where in the diagram yousay secant = ladder hypotenuse, but in the embedded calculator you say % ofhyp. where hyp. means the radius. So, the hypotenuse is the secant or theradius?



64. kalid says:Hi Aurelio, really glad it was helpful.

Good question on the meaning of the hypotenuse. In my head, I see the hypotenuseas the longest side of the triangle.

When were in the dome, the hypotenuse [longest side] touches the dome, and isequal to the radius. When we have the secant, were drawing a line all the way tothe wall [which extends outside the dome]. In this case, the secant is written interms of the hypotenuse, such as 2.5 or 10. [So the secant is 250% or 1000% largerthan the original hypotenuse which is stuck inside the dome. This is also the radiusof the dome.]

65. Aurelio Jargas (http://aurelio.net) says:Ok, now I got it! Youre the best Kalid, thanks for the detailed (and very quick)answer. BTW, youve just sold another book

66. binnoy (http://visualizingmathsandphysics.blogspot.in/) says:Thank You KalidBinnoyhttp://visualizingmathsandphysics.blogspot.in/(http://visualizingmathsandphysics.blogspot.in/)

67. Anonymous says: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 thetheatre ! well not everyone !!

to me you seem to measure water by Grains size !!



68. Theophilus says:Anonymous or what do you call yourself, we know you are a genius but keep it toyourself. We are satisfied with what Khalid is givining. If you so much know trig asyou claimed, why have we not heard about you uptil now? Are you not aware thatpeople dont know mathematics? Are you telling me that that the traditionalteaching 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.

69. Harish Dobhal says:Good reply Theophilus. He or She is Anonymous for some reason and we got it

70. Harish Dobhal says:My guess is he or she is either one of those teachers who have been pursuing theirstudents to memorize trig formulae OR one of those purists who make it their goalof life to keep things as they were and so are blind to actual logic.

71. Bhashit (http://www.bhashitparikh.com) says: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 lastweek during that time. Now, today, during one of those hours, I was amazed to findthat the explanations were still clear in my brain. I mean, I can really visualize thesin/cos etc. Thanks a lot.

72. Fahim says:



Hi Kalid,Longtime lurker of your site, admirer of your ability to intuit.I appreciate exampleslike the hellish voyage on the appendix. I found your site way after leaving collegeand things are finally becoming clear to me. Thanks for your passion.

73. dingo says:Are you a wizard?

74. Usha says:Identifying pause and digest moments.. there are several:11 min video broken down0 Motiv/intro0:38 Trig as anatomy what are the ratios2:00 Analogy of the Dome, Screen, Distance to Screen3:27 Think of in terms of percents4:46 Projecting on a wall6:17 Ramp to the ceiling7:25 Connections: Putting it all together8:38 Facts or relationships we get10:05 Wall or Ceiling firstFor those learning this, realizing utility of pausing at particular moments could behelpful.

Great video!Usha

75. kalid says:Wow, thank you Usha, great summary!



76. Leo says:Kalid:

The fact that trig relations are percentages is a very good insight! I mustcompliment you.

I would have a modest(?) proposal: I am of an older generation, did my studies inthe 50s. We had a visual tool always at our side, an analogue calculator, a thingcalled a SLIDE RULE. It is based on logarithms, and you multiply on it by SLIDINGone log number line scale against another. My big slide rule has trig functions andexponential functions in the form of number lines that refer to the basic log scale.Its all out there in front, right before your eyes. I think my generation and thosebefore 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 becomenatural if you work with a slide rule. And: you have to keep orders of magnitudestraight in your head. A slide rule user must always know where they are in acalculation. You might make little mistakes in the 3rd or 4th significant digit, butnever a BIG electronic calculator type of mistake.

I have never stopped using a slide rule and never really graduated to electroniccalculators. I think it is worth anybodys while who is interested in an intuitive feelfor math to look into slide rules. Their are lots of them floating around and the webhas lots of info on how to use them.

77. kalid says:Hi Leo, thanks for the comment, glad the percentage analogy worked!

The slide rule is a really interesting device (Ive only used it in online simulations,never in real life) but I think theres cool insights within as well. Having a tangiblerepresentation of what logs are doing is another way to build an intuition, might bea fun article!



78. Leo says:Kalid:

One might get a taste by using an online simulation, but there is no substitute forholding one your hands. They are precision instruments, the final product of 300years of development and evolution. And they were expansive. My high school sliderule cost $55 in 2014 dollars, and my big scientific-engineering one was $250. Bothare in working order today after 55-60 years. Back then, a person bought usuallyone high quality one in a lifetime. The German-American rocket engineer Wernervon Braun worked on the Apollo moon shots with one he bought in the early1920s.

(I have never replaced a set of batteries)

I am not suggesting one really learns to use one seriously. With all the relationsthere in front of you, they are great objects for meditations on mathematics. Mostof your insights in this site could be arrived at by figuring out precisely how a sliderule works. If a person uses a slide rule a lot, you can estimate the result of acomputation by closing your eyes, picturing a slide rule, and operating it in yourimagination. Calculators are of course remarkable, but they are black boxes.Nobody knows whats going on inside. The result pops out magically. They arehighly anti-intuitive.

Get your hands on a physical slide rule. Beware: they cast a spell.

I am working my way through your entire site. Great stuff, new little things at everypoint.

79. Sarah says:



Thank you! This helps a lot! I am on winter break, and am studying math on myown to understand it instead of just memorizing it. Ive had two physics classesand up to Calculus I but I am very weak at Trig because I have no idea, it justdoesnt click. Calculus was way easier, but trig popped up everywhere in it and inphysics too and I was just at its mercy! The diagram was the best I alwayswondered where the other identies were located on the triangles/ circle dome. themost useful diagram next to the original unit circle Ive ever come across!

80. kalid says:@Leo: Thanks for the info, glad youre enjoying the site :).

@Sarah: Awesome to hear! Trig bugged me for so long as well until I found a wayto have the relationships click.

81. CAWALING JESRYL BANAYBANAY (http://FACEBOOK.COM) says:it is so great for those are students wanting to know or to understand easily for thissad topic, just scroll this website for more quaries,..

82. Umar danish says:Salam wa alay kum .Mr. Khalid and very very thanx for providing such sites Ithink this is the best site for math seekers.

83. Pat Moss says:Hi.This posting is great! Makes it a lot easier to understand. I knew how to doTrigonometry but realized just how tricky it was when I had to try & explain it tomy teenage son. This certainly made it easier. Thank you.



84. michael edwards says:I never took trig now I am curious and familiar with operations that apply.

85. Andy B says:Just a small tip for anyone out there using Google as a calculator for any of thesefunctions, beware that it returns results in radians. I was puzzled for a few minuteson Kalids example to solve for sine of angle x until I realised the answer Googlewas computing for arcsin(.60) [0.64] was in radians. Performing a conversion fromthat to degrees (by multiplying the radian value by 180/) yielded the expectedresult of 36.9 degrees. Happy Math!

86. Ghostrecon says:I sent the question about the tangent line never reaching x-axis.

87. Ty says:I truly thank God for you. Thanks so much!

88. Jim says:I could always work with the formulas, but had a hard time visualizing them andunderstanding them intuitively. This really helped, thank you!

89. nick says:What happens at x = /4?

90. Ayubi says:Hey Khalid, its really nice to see how you demystify many of the concepts that welearned in high school. Ive always tried to get some intuition but failed so far so Igave up on this quest. Its refreshing to see how youve been able to make it work.



However, I would love to see how you reasoned your way through to getting these 3similar triangles all stacked up on top of one another. Can you please post anarticle or a reply to this comment? Would love to see more of your work!

91. Geraint says:Thanks for the article its great! One question though, is there a reason why thetangent is always vertical??

92. kalid says:Hi Geraint, the tangent doesnt have to be vertical, but its lined up that way forsimplicity in the diagram. See the section of the article Appendix: The OriginalDefinition Of Tangent.

93. kalid says:Hi Ayubi, glad you enjoyed it! It might be a fun article, but I essentially look forconnections between things wherever I can. All the trig functions seem to havesimilar relationships (something^2 + something^2 = something^2) which fits thePythagorean model. Seeing everything as percentages seemed to help clarify aswell (tan is just another percentage, except it can go to infinity). A lot of it is trialand error and a belief that things can be simple if we look at it the right way (and itmay take a lot of time before it jumps out at us).

94. Aaron Brown says:WowI feel like my foot has been itching since high school, and it took me 18 yearsuntil I was finally able to take off my shoe and scratch it. I cant believe how muchsense this post makes. Kudos, Kalid, for making this subject so easy andunderstandable! If only all high school teachers everywhere would watch yourvideos.



LaTeX: $$e=mc^2$$

95. kalid says:Thanks Aaron, glad it helped!

96. Matthew says:*Click*After 25 years, thank you oh so very much.

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In This Series

1. How To Learn Trigonometry Intuitively

2. Intuition For The Law Of Sines (http://betterexplained.com/articles/law-of-sines/)

3. Intuition For The Law Of Cosines (http://betterexplained.com/articles/law-of-cosines/)

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