Intuitive Understanding of Sine Waves

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Sine waves confused me. Yes, I can mumble "SOH CAH TOA" and draw lines within triangles. But what does it mean?

I was stuck thinking sine had to be extracted from other shapes. A quick analogy:

You: Geometry is about shapes, lines, and so on.

Alien: Oh? Can you show me a line?

You (looking around): Uh... see that brick, there? A line is one edge of that brick.

Alien: So lines are part of a shape?

You: Sort of. Yes, most shapes have lines in them. But a line is a basic concept on its own: a beam of light, a direct route on a map, or even a vector in many dimensions. You see--

Alien: Lines come from bricks. Bricks bricks bricks.

The frustration! Because sine is introduced with angles and circles, my brain thinks "Sine comes from circles. Circles circles circles."

No more. In a sentence: Sine is a natural sway, the epitome of smoothness: it makes circles "circular" in the same way lines make squares "square".

Let's build our intuition by seeing sine as its own shape, and then understand how it fits into circles and the like. Onward!

Sine vs Lines

Remember to separate an idea from an example: squares are examples of lines. Sine clicked when it became its own idea, not "part of a circle."

Let's observe sine in a simulator (Email readers, you may need to open the article directly):

Hubert will give the tour:

  • Click start. Go, Hubert go! Notice that smooth back and forth motion? That's Hubert, but more importantly (sorry Hubert), that's sine! It's natural, the way springs bounce, pendulums swing, strings vibrate... and many things move.
  • Change "vertical" to "linear". Big difference -- see how the motion gets constant and robotic, like a game of pong?

Let's explore the differences with video:

  • Linear motion is constant: we go a set speed and turn around instantly. It's the unnatural motion in the robot dance (notice the linear bounce with no slowdown at 0:07, the strobing effect at :38).

  • Sine changes its speed: it starts fast, slows down, stops, and speeds up again. It's the enchanting smoothness in liquid dancing (human sine wave at 0:12 and 0:23, natural bounce at :47).

Unfortunately, textbooks don't show sine with animations or dancing. No, they prefer to introduce sine with a timeline (try setting "horizontal" to "timeline"):

(source)

Egads. This is the schematic diagram we've always been shown. Does it give you the feeling of sine? Not any more than a skeleton portrays the agility of a cat. Let's watch sine move and then chart its course.

The Unavoidable Circle

Circles have sine. Yes. But seeing the sine inside a circle is like getting the eggs back out of the omelette. It's all mixed together!

Let's take it slow. In the simulation, set Hubert to vertical:none and horizontal: sine*. See him wiggle sideways? That's the motion of sine. There's a small tweak: normally sine starts the cycle at the neutral midpoint and races to the max. This time, we start at the max and fall towards the midpoint. Sine that "starts at the max" is called cosine, and it's just a version of sine (like a horizontal line is a version of a vertical line).

Ok. Time for both sine waves: put vertical as "sine" and horizontal as "sine*". And... we have a circle!

A horizontal and vertical "spring" combine to give circular motion. Most textbooks draw the circle and try to extract the sine, but I prefer to build up: start with pure horizontal or vertical motion and add in the other.

Quick Q & A

A few insights I missed when first learning sine:

Sine really is 1-dimensional

Sine wiggles in one dimension. Really. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave.

Circles are an example of two sine waves

Circles and squares are a combination of basic components (sines and lines). But circles aren't the origin of sines any more than squares are the root cause of lines.

What do the values of sine mean?

Sine cycles between -1 and 1. It starts at 0, grows to 1.0 (max), dives to -1.0 (min) and returns to neutral. I also see sine like a percentage, from 100% (full steam ahead) to -100% (full retreat).

What's is the input 'x' in sin(x)?

Tricky question. Sine is a cycle and x, the input, is how far along we are in the cycle.

Let's look at lines:

  • You're traveling on a square. Each side takes 10 seconds.
  • After 1 second, you are 10% complete on that side
  • After 5 seconds, you are 50% complete
  • After 10 seconds, you finished the side

Linear motion has few surprises. Now for sine (focusing on the "0 to max" cycle):

  • We're traveling on a sine wave, from 0 (neutral) to 1.0 (max). This portion takes 10 seconds.
  • After 5 seconds we are... 70% complete! Sine rockets out of the gate and slows down. Most of the gains are in the first 5 seconds
  • It takes 5 more seconds to get from 70% to 100%. And going from 98% to 100% takes almost a full second!

Despite our initial speed, sine slows so we gently kiss the max value before turning around. This smoothness makes sine, sine.

For the geeks: Press "show stats" in the simulation. You'll see the percent complete of the total cycle, mini-cycle (0 to 1.0), and the value attained so far. Stop, step through, and switch between linear and sine motion to see the values.

Quick quiz: What's further along, 10% of a linear cycle, or 10% of a sine cycle? Sine. Remember, it barrels out of the gate at max speed. By the time sine hits 50% of the cycle, it's moving at the average speed of linear cycle, and beyond that, it goes slower (until it reaches the max and turns around).

So x is the 'amount of your cycle'. What's the cycle?

It depends on the context.

  • Basic trig: 'x' is degrees, and a full cycle is 360 degrees
  • Advanced trig: 'x' is radians (they are more natural!), and a full cycle is going around the unit circle (2*pi radians)

Play with values of x here:

But again, cycles depend on circles! Can we escape their tyranny?

Pi without Pictures

Imagine a sightless alien who only notices shades of light and dark. Could you describe pi to him? It's hard to flicker the idea of a circle's circumference, right?

Let's step back a bit. Sine is a repeating pattern, which means it must... repeat! It goes from 0, to 1, to 0, to -1, to 0, and so on.

Let's define pi as the time sine takes from 0 to 1 and back to 0. Whoa! Now we're using pi without a circle too! Pi is a concept that just happens to show up in circles:

  • Sine is a gentle back and forth rocking
  • Pi is the time from neutral to max and back to neutral
  • n * Pi (0 * Pi, 1 * pi, 2 * pi, and so on) are the times you are at neutral
  • 2 * Pi, 4 * pi, 6 * pi, etc. are full cycles

Aha! That is why pi appears in so many formulas! Pi doesn't "belong" to circles any more than 0 and 1 do -- pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!

Question: If pi is half of a natural cycle, why does it go on forever (i.e., irrational)?

Can I answer a question with a question? Why does the diagonal of a "unit square" have length sqrt(2), which also goes on forever?

But yes, I realize it's philosophically inconvenient when nature behaves randomly. I don't have a good intuition.

How fast is sine?

I've been tricky. Previously, I said "imagine it takes sine 10 seconds from 0 to max". And now it's pi seconds from 0 to max back to 0? What gives?

  • sin(x) is the default, off-the-shelf sine wave, that indeed takes pi units of time from 0 to max to 0 (or 2*pi for a complete cycle)
  • sin(2x) is a wave that moves twice as fast
  • sin(x/2) is a wave that moves twice as slow

So, we use sin(n*x) to get a sine wave cycling as fast as we need. Often, the phrase "sine wave" is referencing the general shape and not a specific speed.

Part 2: Understanding the definitions of sine

That's a brainful -- take a break if you need it. Hopefully, sine is emerging as its own pattern. Now let's develop our intuition by seeing how common definitions of sine connect.

Definition 1: The height of a triangle / circle!

Sine was first found in triangles. You may remember "SOH CAH TOA" as a mnemonic

  • SOH: Sine is Opposite / Hypotenuse
  • CAH: Cosine is Adjacent / Hypotenuse
  • TOA: Tangent is Opposite / Adjacent

For a right triangle with angle x, sin(x) is the length of the opposite side divided by the hypotenuse. If we make the hypotenuse 1, we can simplify to:

  • Sine = Opposite
  • Cosine = Adjacent

And with more cleverness, we can draw our triangles with hypotenuse 1 in a circle with radius 1:

Voila! A circle containing all possible right triangles (since they can be scaled up using similarity). For example:

  • sin(45) = .707
  • Lay down a 10-foot pole and raise it 45 degrees. It is 10 * sin(45) = 7.07 feet off the ground
  • An 8-foot pole would be 8 * sin(45) = 5.65 feet

These direct manipulations are great for construction (the pyramids won't calculate themselves). Unfortunately, after thousands of years we start thinking the meaning of sine is the height of a triangle. No no, it's a shape that shows up in circles (and triangles).

Realistically, for many problems we go into "geometry mode" and start thinking "sine = height" to speed through things. That's fine -- just don't get stuck there.

Definition 2: The infinite series

I've avoided the elephant in the room: how in blazes do we actually calculate sine!? Is my calculator drawing a circle and measuring it?

Glad to rile you up. Here's the circle-less secret of sine:

Sine is acceleration opposite to your current position

Using our bank account metaphor: Imagine a perverse boss who gives you a raise the exact opposite of your current bank account! If you have $50 in the bank, then your raise next week is -$50. Of course, your income might be $75/week, so you'll still be earning some money ($75 - $50 for that week), but eventually your balance will decrease as the "raises" overpower your income.

But never fear! Once your account hits negative (say you're at -$50), then your boss gives a legit $50/week raise. Again, your income might be negative, but eventually the raises will overpower it.

This constant pull towards the center keeps the cycle going: when you rise up, the "pull" conspires to pull you in again. It also explains why neutral is the max speed for sine: If you are at the max, you begin falling and accumulating more and more "negative raises" as you plummet. As you pass through then neutral point you are feeling all the negative raises possible (once you cross, you'll start getting positive raises and slowing down).

By the way: since sine is acceleration opposite to your current position, and a circle is made up of a horizontal and vertical sine... you got it! Circular motion can be described as "a constant pull opposite your current position, towards your horizontal and vertical center".

Geeking Out With Calculus

Let's describe sine with calculus. Like e, we can break sine into smaller effects:

  • Start at 0 and grow at unit speed
  • At every instant, get pulled back by negative acceleration

How should we think about this? See how each effect above changes our distance from center:

  • Our initial kick increases distance linearly: y (distance from center) = x (time taken)
  • At any moment, we feel a restoring force of -x. We integrate twice to turn negative acceleration into distance:

\displaystyle{ \iint -x = \frac{-x^3}{3!} }

Seeing how acceleration impacts distance is like seeing how a raise hits your bank account. The "raise" must change your income, and your income changes your bank account (two integrals "up the chain").

So, after "x" seconds we might guess that sine is "x" (initial impulse) minus x^3/3! (effect of the acceleration):

Something's wrong -- sine doesn't nosedive! With e, we saw that "interest earns interest" and sine is similar. The "restoring force" changes our distance by -x^3/3!, which creates another restoring force to consider. Consider a spring: the pull that yanks you down goes too far, which shoots you downward and creates another pull to bring you up (which again goes too far). Springs are crazy!

We need to consider every restoring force:

  • y = x is our initial motion, which creates a restoring force of impact:
  • y = -x^3/3!, which creates a restoring force of impact:
  • y = x^5/5!, which creates a restoring force of impact:
  • y = -x^7/7! which creates a restoring force of impact...

Just like e, sine can be described with an infinite equation:

\displaystyle{sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ... }

I saw this formula a lot, but it only clicked when I saw sine as a combination of an initial impulse and restoring forces. The initial push (y = x, going positive) is eventually overcome by a restoring force (which pulls us negative), which is overpowered by its own restoring force (which pulls us positive), and so on.

A few fun notes:

  • Consider the "restoring force" like "positive or negative interest". This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to learn here :).
  • For small amounts, "y = x" is a good guess for sine. We just take the initial impulse and ignore any restoring forces.

The Calculus of Cosine

Cosine is just a shifted sine, and is fun (yes!) now that we understand sine:

  • Sine: Start at 0, initial impulse of y = x (100%)
  • Cosine: Start at 1, no initial impulse

So cosine just starts off... sitting there at 1. We let the restoring force do the work:

\displaystyle{y = 1 - \frac{x^2}{2!}}

Again, we integrate -1 twice to get -x^2/2!. But this kicks off another restoring force, which kicks off another, and before you know it:

\displaystyle{cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + ...}

Definition 3: The differential equation

We've described sine's behavior with specific equations. A more succinct way (equation):

\displaystyle{y'' = -y}

This beauty says:

  • Our current position is y
  • Our acceleration (2nd derivative, or y'') is the opposite of our current position (-y)

Both sine and cosine make this true. I first hated this definition; it's so divorced from a visualization. I didn't realize it described the essence of sine, "acceleration opposite your position".

And remember how sine and e are connected? Well, e^x can be be described by (equation):

\displaystyle{y'' = y}

The same equation with a positive sign ("acceleration equal to your position")! When sine is "the height of a circle" it's really hard to make the connection to e.

One of my great mathematical regrets is not learning differential equations. But I want to, and I suspect having an intuition for sine and e will be crucial.

Summing it up

The goal is to move sine from some mathematical trivia ("part of a circle") to its own shape:

  • Sine is a smooth, swaying motion between min (-1) and max (1). Mathematically, you're accelerating opposite your position. This "negative interest" keeps sine rocking forever.
  • Sine happens to appear in circles and triangles (and springs, pendulums, vibrations, sound...)
  • Pi is the time from neutral to neutral in sin(x). It doesn't "belong" to circles any more than 0 and 1 do.

Let sine enter your mental toolbox (Hrm, I need a formula to make smooth changes...). Eventually, we'll understand the foundations intuitively (e, pi, radians, imaginaries, sine...) and they can be mixed into a scrumptious math salad. Enjoy!

Other Posts In This Series

  1. A Visual, Intuitive Guide to Imaginary Numbers
  2. Intuitive Guide to Angles, Degrees and Radians
  3. Intuitive Arithmetic With Complex Numbers
  4. Understanding Why Complex Multiplication Works
  5. Intuitive Understanding Of Euler's Formula
  6. An Interactive Guide To The Fourier Transform
  7. Intuitive Understanding of Sine Waves
Kalid Azad loves sharing Aha! moments. BetterExplained is dedicated to learning with intuition, not memorization, and is honored to serve 250k readers monthly.

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99 Comments

  1. Excellent work! Thank you.

    I particularly enjoyed having the infinite series model click intuitively, and seeing that the unit circle contained all possible right triangles. Why, yes, yes it does!

  2. Brilliant. I must agree with Erich, the infinite series visualization is wonderfully intuitive.

  3. @D-POWER: Awesome ;)

    @Erich/Anonymous: Thanks for letting me know what made it click! I’m working on an idea to make it easier to share these types of aha moments.

  4. Your graph “Better Models of Sine” illustrating the successive series approximations of sine has an error: it indicates that sin(x) = x – x³/3! + x⁵/5! – x⁷/y! + x⁹/9! – x⁹/9! + x¹¹/11! …, which includes x⁹ twice rather than once. With as many terms, it should be sin(x) = x – x³/3! + x⁵/5! – x⁷/y! + x⁹/9! – x¹¹/11! + x¹³/13! …

  5. @Anon1: Thank you!

    @Anon2: Glad it helped — sine has bugged me for so long.

    @Anon3: Love those pictures! Our brains need both :).

  6. @Anonymous: That’s just the names we’ve given to those ratios, like saying perimeter = 4 * side [in a square].

    But as it turns out, sine isn’t limited to triangles — that is just the first place it was noticed.

  7. I haven’t made the connection between sine as an idea and why the ratios in SOHCAHTOA are what they are. Am I making sense?

  8. Wow, thanks once again Kalid. Your explanatiosn are truly wonderful, just how do you come to such a level of knowledge and how do you manage to explain it so easily?
    I wish you were my teacher.
    Every article is just magic, please keep writing it’s a real relief every time you release another article.

  9. Thank you!

    That was a fantastic lesson. Since I left school I’ve come back to math every few years to try and remember everything I’d forgotten. The best feeling in the world (yep, even better than *that* one…) is the “Eureka!” moment when everything just makes sense. Your article gave me two of those, from watching Hubert move in his circle and from seeing the derivitave definition of sine and how it related to e. You have a gift for teaching and writing, thank you for sharing it.

  10. @Anonymous: I put an answer at http://www.reddit.com/r/learnmath/comments/guyik/why_do_the_ratios_in_sohcahtoa_work/, let me know if it helps!

    @nschoe: Thanks for dropping by! I appreciate the kind words — I don’t think I really understand that much, it’s more my lack of understanding/satisfaction which drives me to seek simpler explanations. The notion that sine is this cyclical wave that we all see just didn’t click deeply with me, I needed something deeper. Many ideas are like that (e, imaginaries, etc.) so I start trying to find analogies that might fit better :).

    I’ll definitely keep writing, appreciate the support!

    @loimprevisto: You’re welcome! You got it, those Eureka moments are so incredibly fulfilling. It’s what I strive for when writing, I just want to share what clicked hoping it clicks for other people too. Thanks for sharing what aspects helped (Hubert / derivative definition), I have a project in the works to make these insight exchanges easier & more community driven :).

  11. Since my engineering studies I always liked Euler’s formula, connecting sine and cosine to the unit circle in the complex number plane. That’s what Hubert’s sine-sine setting reminded me of.

  12. I’ve been reading these for a while, I have to say I think this is the best one yet. We did Taylor series a month or so ago in my Calc class, the end of this article aided my comprehension a whole lot more than any of the class work ever did. Keep ‘em coming, please!

  13. @Joe: Thanks — this was one of the longer ones to write so glad it was helpful… I’ll keep cranking :)

  14. You are a master. At some point I would love to hear your take on why the pattern that emerges in transformation matrices
    cos -sin
    sin cos
    changes for rotations about the y axis. For now I”m feeling hungry for a salad….

  15. @mark: Thanks — transformations matrices would be a fun addition.

    @iheartcomputers: Exactly! I was the same way, I thought I understood them too :).

  16. Hi Kalid, great article and great site! But I haven’t understand a thing: the opposite acceleration of x is the double integral of (-x). So we get that, just a moment after the beginning, sin(x) = x – x^3/3!. That’s fine.
    Now we have another opposite acceleration so we have to integrate twice -(x – x^3/3!), don’t we? we then get an acceleration of -x^3/3! + x^5/5!. Summing this to the previous result of sin(x) we get sin(x) = x – 2*x^3/3! + x^5/5!. But this is obviously wrong, as the series has -1*x^3/3! instead of -2*x^3/3!.
    It seems like we don’t have to integrate -(x – x^3/3!) but only x^3/3!; Why? if the acceleration must be the opposite of the current value, I expect to integrate all of its members.
    Thank you very much

  17. @simone: Great question! You might take a look at the diagrams here:

    http://betterexplained.com/articles/an-intuitive-guide-to-exponential-functions-e/

    Basically, each motion (-x) creates interest (-x^3/3!), which creates s interest (+x^5/5!), which creates interest…

    My intuitive understanding is that the initial motion (-x) begins a chain reaction, but the chain goes forward — it doesn’t pull back and change the original. I.e., the interest doesn’t go back and change the original… it just generates its own interest (this might be more clear in the diagrams on that article… Mr. Blue creates Mr. Green who creates Mr. Red… Mr. Blue doesn’t even know about Mr. Red!).

  18. I struggled to teach sine to my friend, but now I realize that I didn’t understand it myself. Not intuitively anyway. Can’t wait for your next post!

  19. @C: Thanks! Yep, often times I don’t get an intuition until visiting the topic a 2nd (or 3rd) time :).

  20. Kalid, you are god of mathematical explanation.I have been visiting this site and I will always visit it.So many things I was taught in college is just making sense now. Thank you!

  21. Your website is brilliant. I am 37 and have always struggled with mathematics. Reading your site makes me feel like I suffered a form of child abuse the way I was taught at school.

    I left high school with the impression that sine “came from” triangles. University classes left me none the wiser. It was only when I bought and read books which were written fairly well that I understood sine to come from circles. It was an epiphany. I was in my mid 20s.

    Now you give me another epiphany, that circles come from sine. Brilliant, but 20 years late! I wish you had been around all those years ago.

    Keep it up.

  22. “pi is about sine returning to center! A circle is an example of a shape that repeats and returns to center every 2*pi units. But springs, vibrations, etc. return to center after pi too!”

    Even Feynman never figured this out:

    “About a half year later, I found another book which gave the inductance of round coils and square coils, and there were other pi’s in those formulas. I began to think about it again, and I realized that the pi did not come from the circular coils. I understand it better now; but in my heart I still don’t know where that circle is, where that pi comes from.”

    http://www.fotuva.org/feynman/what_is_science.html

  23. @Matthew: Hrm, I think the website I linked to may have been down for a bit — it should be back in the article now. Thanks for the note though!

    @Peter: Wow, thanks for the reference! The “pi must be about circles” mantra has been pounded into all of us for a long time :).

  24. @Theo: Wow, thanks for the kind words! I’m basically doing the same as you — going back to relearn what I thought I learned in college :).

    @Michael: Thanks for the kind words and encouragement! There are so many misconceptions that I’m only beginning to unravel (this whole sine business only started clearing up in the last few months). I love that epiphany feeling. Sorry for the delay in reply, I was on vacation when these earlier comments were posted.

    @Jay: You’ll never look at sine the same again :).

    @Zack: Don’t know Chinese unfortunately… Google translate?

  25. yeah. thx for great articles. I hope I can learn more about math, because i will study informatics after summer.

  26. You are awesome!!! Seeing sine as motion and not part of a static geometric diagram is so new! I’ve been really curious about how to understand math in different ways. I see at uni that the concepts learned in school are gussied up in different disguises, based on the discipline.

    I just had a question to clarify what you said about pi as being a notion of time. I didn’t quite understand, given that I’ve always thought of pi as distance. But this is sort of a two-sides of a coin thing…do you mean that distance=rate x time, and just assume that the unit for the rate of sine to go from neutral to neutral is pi/seconds? So that the units cancel out and thus distance is equal to time?

    Also, could you just briefly clarify how you got the successive terms in “The Better Models of Sine” section? x is the initial impulse, and I think I follow your double integration of the opposite position, but then where do you get x5/5! and so forth?

  27. @Ashley: Thanks for the note! Yep, a key to math is trying to see it from different angles — some click better than others.

    1) Yep, pi is like time if you assume rate = “1 unit per second” (or really, 1 unit per unit time). So then you have distance = rate * time, or distance = time.

    If we’re talking about radians (distance traveled along the outside of the circle) then if we go for 2*pi units of time, we’ve traveled a distance of 2*pi and we’re all the way around. If you see sine as a “living, moving” process, the 2*pi is how often the process is back to its initial position. pi by itself is the time from neutral to neutral (middle-top-middle or middle-bottom-middle).

    2) Great question. With e, we’ve seen that our “interest earns interest”. In the same way, the “restoring force” of sine creates its own restoring force, which creates another restoring force… and so on.

    The thing is, these happen _simultaneously_, we don’t need to wait for the first force to happen (or maybe another way, it happens infinitely quickly so we can’t tell). So when we compute sine, we have to account for as many restoring forces as possible to be accurate.

    So, our original force is x (call this A). This creates a restoring force B (pulling us to center) of x^3/3! (the double integral of x). That creates force C which tries to balance the pull center with a push away (double integral of x^3/3! is x^5/5!).

    In this way, A creates B, which creates C, which creates D… to infinity :). Actually, each restoring force is in the opposite direction (x has restoring force -x^3/3! because the rule for sine is your acceleration, your second derivative, is the opposite your current position).

    Hope this helps!

  28. Woah! This is awesome! I can’t believe I found this website yesterday and I was like “Too bad I didn’t find this website some time before my exam, would’ve made my preparations much more fun and versatile”. But then I go to the exam, and it turns out I got the wrong date- it’s actually tomorrow. So I have this whole day free to read a bunch of your articles. They make me all excited about math again! Yeepeee! :D
    Thank you sir, you are a blessing!

  29. Correction: sine is bounded between -1 and 1, inclusively, if the argument is a real number. In fact, there are (infinitely many) complex solutions for sin(x) = 2. For example,
    sin(π/2 + i*ln(2 + √3)) = 2

  30. @Christian: Although it is true that sine takes on values outside of [-1,1] for complex arguments – and in fact takes on *all* complex values (more generally, this is true, with the exception of at most 1 value, for *any* non-constant analytic complex functions, by Picard’s theorem http://en.wikipedia.org/wiki/Picard_theorem), I would argue that this critique is misplaced here as it refers to a mathematical extension of the definition of sine to the complex plane, a space of consideration that was never once intended to be mentioned here. Sine was originally constructed to carry a certain *meaning* – to represent the path of harmonic motion (modulo amplitude) (which, by definition, doesn’t “leave” its interval of oscillation) and parameterize the unit circle (which, again by definition, cannot have a distance from the center greater than 1).

    This lesson aimed to explain the origin of, intuition behind, and exposition of: the sine wave, not the general rigorous properties of the sine function over the complex field (which belongs in a class on complex analysis, not in a blog post on high school algebra).

    In addition, @Kalid only mentions these “bounds” (he, in fact, never uses this more formal term that carried the very baggage that you intended to dispose of) within the language of “moving” and “swaying” – these already imply that we are on the real line: intuitively this is because motion implies time (or at the very least – 1 dimensional travel) which, for all intents and purposes, is “real” (as C is 2 dimensional – we cannot “trace” across the entire vector field of sine over C).

    Often concepts originally defined to mean something spacial/physical get generalized with mathematical language, and along the way lose some of the properties that originally motivated their definition in the first place – this does not mean that the explanation (or even conception) of them in those original terms is “incorrect” – there is no “one true sine function” but rather a general notion of sine with distinct appropriate definitions for different contexts. So, correction: your comment was just an interesting addendum.

  31. Sine is the point that rotates around another fixed point which causes motion in a linear fashion on a line bisecting the fixed point? Like the arm affixed to the wheels of a steam locomotive running along a track?

    I don’t get it still… Worse yet is how, from this, am I to make heads or tails about trig identities? They seem to be everywhere…

  32. @Anonymous: Great question. I see sine as a general sway back and forth. We can notice this sway on a circle by realizing the height of a point sways up and down if we just look at that axis. It’s a bit like the arm on the wheels on a locomotive, yep. (But again, that’s just an example — sine is a general concept which shows up everywhere! Some of the confusion is around the definition of sine (it’s a sway!) vs. examples (it shows up in circles, and pendulums, and so on).

    Trig identities are another beast entirely. Basically, it turns out that one “sway” (sine) can have relationships to other sways (cosine), for example sin^2 + cos^2 = 1. I’d like to cover this more!

  33. How fast is sine?
    I have a question and a comment.

    I get what you are saying about sin(x) going from 0 to max in pi units of time. But that is only true when you are using radians as your measure of time. And radians are defined (I think) by the angle you can get around a circle by traveling along the circumference a distance of 1 radius.

    So is it really fair to say that pi doesn’t belong to circles? Without radians, there’s no pi. Without circles, there are no radians. So without circles, there’s no pi. Is that not so?

    Also, you say: Circular motion can be described as “a constant pull opposite your current position, towards your horizontal and vertical center”. I don’t think that’s what you meant to say, is it? Didn’t you mean to say that circular motion is a constant pull perpendicular to current velocity? A planet orbiting on a circular path experiences a force and an acceleration that is always perpendicular to its path (and has the same strength throughout the orbit, else the orbit is an elipse).

    Likewise, I think to be precise you don’t want to say sine is acceleration opposite to your curent position. Sine (one dimensional) is acceleration toward the origin, in proportion to the distance from the origin.

  34. I don’t get how you got this part:

    \displaystyle{ \int \int -x = \frac {-x^{3}} {x!} }

    I guess I need to learn calculus (BTW, when are you planning to start the series you promised), but can you try explaining what’s happening?

  35. So I was seriously taking on trig (apparently it’s a prerequisite for calculus), and I came upon trig and family. 3 sides of a triangle having 6 relationships called Sine, Cosine, Tangent, Cotangent, Secant, Cosecant. Is that it? Is that all there is to trig apart from learning to manipulate them?

    I read your article a couple of times, but I still didn’t get a few things. You said the `x` in `sin(x)` was how far along it was in a wave. But then you said the `x` meant how fast in relation to the ‘normal’ wave which took `π` seconds to return to neutral. What gives?

    And why can’t we calculate all of these by hand? I mean, sine and cosine and all are functions, right? They must have an exact definition. Why can’t we figure them out by hand? Why do we need these ‘log books’? That reminds me. Is there a better way to find logs than tables? Some sort of a approximation method maybe…

  36. g88888888888888 post as usual specially the series explanation is awesome.
    But here comes a question!!!!!!
    From 0 to pi there are only four reactions or reactions to reactions…but shouldnt it be a continuous chain of reaction, i mean there should be infinite terms between 0 to pi.

  37. @Ram: Thanks!

    @mra:

    1) Asking “How fast is sine?” is like saying “How fast is a circle?”. It’s a general shape, which you can traverse as quickly or slowly as you need. By “default” we use radians to measure angle, and get through the entire neutral-max-neutral-low-neutral cycle in 2*pi radians (6.28).

    2) “Is it really fair to say that pi doesn’t belong to circles?”. It depends on your point of view. I could imagine a world where sine was discovered first (from the motion of springs, let’s say), then pi was discovered, and later on, the shape of a circle was discovered.

    3) I think we’re saying the same thing. A pull opposite your current position vector is towards the center, which the direction of the pull. If you are at (1,0) and moving in a circular path, then your position vector is pointing East, your velocity is pointing North, and your acceleration is pointing West [towards the origin].

    @Yatharth: Yep, you’ll need calculus to decipher that :). It’s basically saying “Sine accelerates your opposite of your position (if your position is x, your acceleration is -x).” To find the total distance that this negative acceleration will impact you, you integrate twice, and get -x^3/3!

    Trig is basically the anatomy for circles and triangles. Learning every part of them, how they’re connected, how to find the sizes of one part given a different one.

    To clarify: the variable is how fast you are along in your wave. If I write sin(pi) I mean “I am pi units along in the wave which takes 2*pi units total”, which means I’m at the halfway point. If I write sin(2x), then I am going to travel the wave twice as fast as the regular sin(x) [since I'll be twice as far along for the same x value].

    You can find sine/cosine by hand, but it’s painful. You plug in values of “x” in that infinite equation [but only take as many terms as your sanity can handle]. There are shortcuts for finding logs, sine, etc. by hand but are no longer really used, for obvious reasons. The first log tables took dozens of man-years to make.

    @podAhmad: I’m not really sure what you mean by only 4 reactions or reactions to reactions. There’s an infinite sequence of them, but I only showed a few terms (with … for the continuation) in the equations.

  38. Hey Kalid, thanks for your great and inspiring posts.

    I would like to have a question here. You mentioned that 2*pi is the time it takes to travel back and forth. I wonder if there is any way to deduce this? (i’m thinking of using the relation acceleration y”=-y). Thanks so much :)

  39. A few years ago I took a basic AC/DC class where I remember that we made and measured sine waves on an oscilloscope and my partner and I after we finished our project made this https://sphotos-b.xx.fbcdn.net/hphotos-ash4/292060_548653444059_1236366013_n.jpg Any chance you could tell me what we did cause we could just keep adding them on and our professor was pretty confused. It’s been bugging me for a while now since I can’t find anything similar online.

  40. From atoms that shape spheres, to earth, to the moon, to the sea waves, sines is no other thing than the projection of the circular motion, stable, in decay or increasing, being it light, sound, radio waves, all are sinusoidal, in fact sine math as a two planes is unrealistic since most waves are tridimensional, So the circle is not a matter of geometry, geometry is one of the results of the study of circles. Men invented math not the circle, neither the waves.
    If you love math, what I have always needed is an equation for the linearity of a sphere, because it would explain much of the quantum behaviour. Imagine an sphere, must be an equation that starting from any point it follows a path such that forms a perfect balance sphere. That line may cross itself but if it does you must balance the exact opposite site if there is any at all. If there is no way to do it, then my question stills. What is the balance of a particle all scientific draws as spheres, even ignoring the particles, they translate the same for atoms and talk about electron spin, and I know you know what happens when something unbalance spins. Is it he reason of the quantum behaviour?

  41. Hey, Kalid!

    First of all, thanks for the whole site. It’s really changed my definition of “knowing” something, and it’s given me a new zest for learning!

    I had a question about the pi without pictures part. I’m really interested in pi not being related to circles, but circles being related to pi. Problematically, though, I feel like I could show that sightless alien a perfectly smooth oscillation that had a period that _wasn’t_ related to pi. I feel like a perfectly smooth shade-oscillation with period 2 seconds isn’t any less smooth than the pi version.

    If the smoothness of oscillation defines pi, what makes regular sin(x) any more special than sin(x*pi) for example?

  42. Zeno, there is no pi not being related to circles, since pi is a constant which equals the number of times the length of the radius of ANY circle size, fits in half around the circle.
    pi is the result of having found that relation between the radius and the perimeter of a circle is always the same, That is whythe whole perimeter of a circle = 2*pi*R
    You may draw a series of points in space and calculate its position from an origin based on the sine, and the final figure or equation may not result in a sine wave, then what is sine?
    sine is not related to circles, but to triangles rectangles, where one angle is 90 degrees, there is a constant relation between the 3 sides, csquare = asquare + bsquare the sum of the squares of the shorter side equals the square of the longer side, no matter the size of the triangle
    So the same as pi, that relation in triangles got related to angles in a 360 degrees cycle (read it cycle, not circle, this is a whole turn around something) so you can imagine something doing a whole cycle which might not be the route of a circle (radius variation if you want) so the cycle refers to 360 degrees, and sine refers to a triangle rectangle side relations, and as you can see we are not tallking neither refering to circles which in fact do not participate.

  43. Wow…My mind is blown.No one ever teaches students these things,and they can make a world of a difference.

    Your articles are phenomenal and so is your attitude.

  44. I agree that defining sine in terms of its second derivative, or “acceleration opposite to position”, is an excellent alternative to the triangle and circle definitions, and makes the connection between sin x and \displaystyle{e^x} much clearer.

    Have you looked at the hyperbolic sine function (sinh)? It’s like a half-way house between sin x and \displaystyle{e^x}, and its graph is distinctly wave-like, though a surreal wave! Its first derivative is the hyperbolic cosine (cosh), but its second derivative is itself. Acceleration is equal to position, as in the case of \displaystyle{e^x}.

  45. Thanks Colin! Great point with the hyberbolic versions — I’ve only dabbled in them very briefly. I like that they can be defined in terms of exponentials… they’d be a fun topic to get into :).

  46. Hey Kalid,

    Keep up the great work! It’s really amazing.

    I was wondering if you might be able to offer any intuition as to why sine of an angle is also the ratio of the opposite side to the hypotenuse in a right triangle? I get that right triangles would have consistent ratios for the different side lengths, but how is sine (and cosine, tangent) able to describe these ratios?

  47. @Stephen: Thanks! Great question on the connection between sine and triangles. I actually prefer starting with circles, and then seeing how triangles fit in.

    Let’s say we’re traveling around the unit circle (radius 1). As we go around, we have some height above the x-axis. From this discussion, we can see that a circle is made up of two “sways”, one controlling the vertical position (call that sway sine), and another controlling the horizontal position (call that sway cosine).

    The only difference between the sways is where they start: sine starts neutral, and starts moving up. Cosine starts at its max value, and starts moving towards center. For now, let’s think about sine, our vertical position.

    If we take our spot on the circle and extend a line down, and to the center of the circle, we end up making a right triangle! The height, our sine, is the side “opposite” to the angle, and the line to the center is the hypotenuse. The resulting triangle must be a right triangle, because we drop a line straight down from our current position.

    Now, by an accident of history, we started working with triangles before circles, so we found this “vertical pattern” (sine) in triangles, then built up to circles. But again, I’d prefer to start with circles and work down to the triangles buried inside.

    Another way to put it: the swaying vertical motion of a circle, called sine, can also be seen in the swaying heights of the right triangles formed along our circular path.

    (The last step is to allow for any hypotenuse, not just the unit circle, so we scale by “H” to show that sine is present in any circle no matter how large, similar to how pi is present in any circle, no matter the radius.)

  48. Hi Kalid,

    Thank you so much for your speedy reply. I really appreciate it.

    I really see what you mean when it comes to thinking of sin and cos as ‘swaying’. I think I’m starting to really internalize that way of looking at it, which definitely is helping to develop my intuitive grasp of these functions.

    However, I think my intuition is still at the stage where I begin with the assumption that circles are made up of a perfect balance of two complementary sways as described by sin and cos, and THEN work backwards to intuitively confirm this initial assumption.

    Ideally, I would like my intuition to see how sin and cos connect to circles without first assuming that they do, and then finding different ways to confirm that this is in fact the case.

    For example, your article on e delved into the math in an incredibly intuitive way, to the point where I saw exactly how the general concept of growth fell directly out of the details of the equations. Or with your article on Pythagorean theorem, where you explicitly highlighted the physical areas that were implicitly being added, which accounted for the squared terms in the equation.

    So I’m wondering if you think it’s possible to approach sin and cos the same way, where circles can kind of ‘fall out’ of the math. Because right now I’m at the stage where my intuition is only confirming, rather than deriving. Perhaps if you know any of the history of how these functions were first discovered…that is often very helpful in developing intuition that derives rather than confirms.

    Please let me know if this isn’t making sense and I will try and elaborate more. I’m being a little bit vague because I’m still trying decide if I’m at the point where I need to accept something as empirically true, or if I can reduce my understanding to more fundamental principles.

    Thanks so much for bearing with me, Kalid. I appreciate your work so much

    Best,
    Stephen

  49. Hi Stephen, great comment. To be honest, my intuition for sine/cosine isn’t as strong as the ones for e, Pythagorean Theorem, etc. so I’m a bit in the “discovering” vs “deriving” mode myself.

    I might say something like this: suppose we have a concept of perfectly smooth growth, which is epitomized by e^x.

    If we combine the idea of perfectly smooth growth (e^x) with rotations (imaginary numbers), we get e^ix.

    Intuitively, before graphing anything, we should imagine that e^ix results in something along the lines of “perfectly smooth rotation”.

    What would this shape be? Well, it should be symmetrical (why would it favor one side over the other?). It should embody the essence of rotation, spinning. And so on.

    Pretty soon, we might see that a circle is the shape which satisfies this intuition. Now the question of sine/cosine comes in.

    This circle exists in 2d: if we analyze each dimension independently, it seems like each dimension should be moving perfectly smoothly as well (again, why would one be favored over the other?). We can’t rotate in a single dimension, but whatever motion we have, should be smooth.

    That pattern of motion, the smooth sway in a single dimension, can be called sine. And we can work out that

    e^ix [perfectly smooth rotation] = cos(x) [smooth sway in the horizontal direction] + i*sin(x) [smooth sway in the vertical direction]

    We’ve separated our 2d motion into a combination of two 1d trajectories. Getting into even more nitty gritty, the series expansion of e^ix = series expansion of cos(x) + series expansion of i*sin(x)

    That is… cosine and sine can literally be “factored” out of the combined circular path we see in e^ix.

    Hopefully that helps?

  50. Hi Kalid,

    Thanks again for some more great insights. Very helpful and definitely adding to my understanding of sin.

    So I think I see everything up to the point where we make the leap from rectangular values (x’s and y’s) to polar values (theta). I think the gap in my intuitive understanding of sin stems from not being able to see where the Taylor series of sin comes from.

    I think your article is great at teasing out the corroborative intuition on the Taylor series: given the existence of the series itself, you demonstrate that the alternating positive and negative terms are corrective tugs in opposite directions, which gives rise to the wave appearance. I get that, and I think it’s a great insight.

    But do you think there is a way to draw a diagram with some triangles or circles or something, so we can start with rectangular x’s and y’s, and derive the Taylor series intuitively, which we THEN define to be the function called “sin(x)”? (rather than say the commonly used circular derivation, where the definition of sin and cos is a given, and based on the fact that each function’s derivative is the negative of the other, you come up with a Taylor series to satisfy that condition. That’s a corroborative approach to be)

    I like how you point out that something more ‘fundamental’ is going on with the trig functions that goes beyond circles: periodicity. But at the end of the day, are the Taylor series of these functions capable of being derived without using the visualization of a circle?

    The person who asked this question on math.stackexchange is getting at what I’m looking for:
    http://math.stackexchange.com/questions/185356/rigorous-proof-of-the-taylor-expansions-of-sin-x-and-cos-x

    However, the most up-voted explanation here is, to me, circular and un-intuitive. It employs the definition of sin and cos and some trig identities to show where the Taylor series came from, which is to me unhelpful.

    Let me know if I’m chasing ghosts. Thanks again for your time, Kalid

    Best,
    Stephen

  51. …..in the same way the you very intuitively explained the Taylor series for e. In that case, I could visualize what each term corresponded to.

  52. Firstly, I’d like to thank you for putting in the time & the work to even be able to explain math in this way. It helped me very much in a time of need. :)

    Secondly, I apologize for not going through all your comments to see if this question had already been answered, but could you possibly explain tangent, secant and cosecant wave patterns on a graph? Or if you’ve already answered that somewhere, post the link to the page.

    Much appreciated.

    Cheers.

  53. Your explanation of how SINE originated is brilliant. My husband (has PhD) and I have Master’s degrees and never thought of this explanation until our highschool daughter asked me this. I am very glad that you have this site.

  54. Thanks GV, really glad you enjoyed it! I didn’t start seeing the meaning of Sine until maybe a decade after learning it “officially”.

  55. About 10 years of studding (from the first day sine was introduced to us in high-school) and ‘today’ I actually learned what sine really is. Dear kalid, you are the best teacher ever. I wish others were like you too.

  56. To all the other great examples of periodic motion, may I add the trampoline? In my mind, I’ve been having fun bouncing from pi to pi.

  57. Let me see if I have this right. Sin(x) is a position function. Sin(x)=x at first, a linear relationship, because there is little or no resistance around the neutral position. But then as the distance increases, so does the resistance, and a restoring force must be added to x, which alters the position. This restoring force, -x, is negative acceleration, a pull back towards the neutral position. We know that acceleration is the second derivative of the position function. Thus, to determine the new position, we double integrate -x to arrive at -x^3/3! and add it to the series. However, this being the inexact world of harmonic motion, the new term overshoots the mark and requires a correction. So we then double integrate -x^3/3!, being sure to cancel the negatives first, which gives us x^5/5!, which also overshoots true sine and must be corrected with another term, and so on.

    This is the most dynamic view of both sine and the Taylor series that I’ve ever seen.

  58. Hi Tim! Yes, that’s exactly it. As the restoring force slows you down (changing your distance by -x^3/3!), it means you aren’t going along your original “x” trajectory any more, and should not feel the full brunt of -x^3/3! — you get a bonus of x^5/5!.

    But, that bonus means you’re going further along than you expected, so feel an extra restoring force of -x^7/7!. And the cycle keeps going :). Basically, the longer you want to model sine, the more levels of restoring forces you need to stay accurate [and if we're on a very small timescale, we just assume sin(x) = x and ignore the impact of even the first restoring force].

    Glad you like the perspective, I need to see/visualize things to make sure they’re really clicking.

  59. Your clarity of thought is a gift. I teach college and often draw upon your analogies. Thank you for sharing your insights generously.

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LaTeX: $$e=mc^2$$