How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (/) How To Learn Trigonometry Intuitively by Kalid Azad · 101 comments Tweet 73 Trig mnemonics like SOH-CAH-TOA (http://mathworld.wolfram.com/SOHCAHTOA.html) focus on computations, not concepts: TOA explains the tangent about as well as x2 + y2 = r2 describes a circle. Sure, if you’re 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 the stunning beauty of an abstract ratio. I think you deserve better, and here’s what made trig click for me. Visualize a dome, a wall, and a ceiling Trig functions are percentages to the three shapes http://betterexplained.com/articles/intuitive-trigonometry/ Page 1 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM Motivation: Trig Is Anatomy Imagine Bob The Alien visits Earth to study our species. Without new words, humans are hard to describe: “There’s a sphere at the top, which gets scratched occasionally” or “Two elongated cylinders appear to provide locomotion”. After creating specific terms for anatomy, Bob might jot down typical body proportions (http://en.wikipedia.org/wiki/Body_proportions): The armspan (fingertip to fingertip) is approximately the height A head is 5 eye-widths wide Adults are 8 head-heights tall http://betterexplained.com/articles/intuitive-trigonometry/ Page 2 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM (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 the owner’s height. And head size. And eye width. One fact is linked to a variety of conclusions. Even better, human biology explains human thinking. Tables have legs, organizations have heads, crime bosses have muscle. Our biology offers ready-made analogies that appear in man-made creations. Now the plot twist: you are Bob the alien, studying creatures in math-land! Generic words like “triangle” aren’t overly useful. But labeling sine, cosine, and hypotenuse helps us notice deeper connections. And scholars might study haversine, exsecant and gamsin (http://www.theonion.com/articles/nations-math-teachers- http://betterexplained.com/articles/intuitive-trigonometry/ Page 3 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM introduce-27-new-trig-functi,33804/), like biologists who find a link between your fibia and clavicle. And because triangles show up in circles… …and circles appear in cycles, our triangle terminology helps describe repeating patterns! Trig is the anatomy book for “math-made” objects. If we can find a metaphorical triangle, we’ll get an armada of conclusions for free. Sine/Cosine: The Dome Instead of staring at triangles by themselves, like a caveman frozen in ice, imagine them in a scenario, hunting that mammoth. Pretend you’re in the middle of your dome, about to hang up a movie screen. You point to some angle “x”, and that’s where the screen will hang. http://betterexplained.com/articles/intuitive-trigonometry/ Page 4 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM The angle you point at determines: sine(x) = sin(x) = height of the screen, hanging like a sign cosine(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. It’s at the center, on top of your head, but it’s big dagnabbit. Want the screen the furthest away? Sure. Point straight across, 0 degrees. The screen has “0 height” at this position, and it’s far away, like you asked. The height and distance move in opposite directions: bring the screen closer, and it gets taller. Tip: Trig Values Are Percentages Nobody 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. Let’s say I paid $14 in tax. You have no idea if that’s expensive. But if I say I paid 95% in tax, you know I’m getting ripped off. http://betterexplained.com/articles/intuitive-trigonometry/ Page 5 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM An absolute height isn’t helpful, but if your sine value is .95, I know you’re almost at the top of your dome. Pretty soon you’ll hit the max, then start coming down again. How do we compute the percentage? Simple: divide the current value by the maximum possible (the radius of the dome, aka the hypotenuse). That’s why we’re told “Sine = Opposite / Hypotenuse”. It’s to get a percentage! A better wording is “Sine is your height, as a percentage of the hypotenuse”. (Sine becomes negative if your angle points “underground”. Cosine becomes negative when your angle points backwards.) Let’s simplify the calculation by assuming we’re on the unit circle (radius 1). Now we can 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 out the 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 Cosine R1 x = 30 30 R2 sine 50 % of height R3 cosine 86.60254037844 % of width The growth pattern of sine isn’t an even line. The first 45 degrees cover 70% of the height, and the final 10 degrees (from 80 to 90) only cover 2%. This should make sense: at 0 degrees, you’re moving nearly vertical, but as you get to the top of the dome, your height changes level off. Tangent/Secant: The Wall http://betterexplained.com/articles/intuitive-trigonometry/ Page 6 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM One day your neighbor puts up a wall right next to your dome. Ack, your view! Your resale 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) and figure out: tangent(x) = tan(x) = height of screen on the wall distance 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 “TAN GENTleman” projected on the wall. You climb the ladder, making sure you can “SEE, CAN’T you?”. (Yeah, he’s naked… won’t forget the analogy now, will you?) Let’s notice a few things about tangent, the height of the screen. http://betterexplained.com/articles/intuitive-trigonometry/ Page 7 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM It starts at 0, and goes infinitely high. You can keep pointing higher and higher on the wall, to get an infinitely large screen! (That’ll cost ya.) Tangent is just a bigger version of sine! It’s never smaller, and while sine “tops off” 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 there Secant is always longer than tangent. The leaning ladder used to put up the screen must be longer than the screen itself, right? (At enormous sizes, when the ladder is nearly vertical, they’re close. But secant is always a smidge longer.) 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 distance to the wall (the radius of the dome). Tangent and Secant R1 x = 50 50 R2 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 Ceiling Amazingly enough, your neighbor now decides to build a ceiling on top of your dome, far into the horizon. (What’s 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 angle to build and work out: http://betterexplained.com/articles/intuitive-trigonometry/ Page 8 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM cotangent(x) = cot(x) = how far the ceiling extends before we connect cosecant(x) = csc(x) = how long we walk on the ramp the 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 the ceiling. Bummer. The shortest “ramp” is when you point 90-degrees straight up. The cotangent is 0 (we didn’t move along the ceiling) and the cosecant is 1 (the “ramp length” is at the minimum). Visualize The Connections A short time ago I had zero “intuitive conclusions” about the cosecant. But with the dome/wall/ceiling metaphor, here’s what we see: http://betterexplained.com/articles/intuitive-trigonometry/ Page 9 of 48 How To Learn Trigonometry Intuitively | BetterExplained 9/26/15, 12:19 AM Whoa, it’s the same triangle, just scaled to reach the wall and ceiling.