Intuitive Understanding of Sine Waves

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

This is a companion discussion topic for the original entry at

This was excellent! Well done.

@Uwe: Thanks!

Another great article from master Kalid, I’m really happy :D.

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!

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

@D-POWER: Awesome :wink:

@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.

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! …

@Polyergic: Doh! Great catch – it should be fixed now :).

Wow! Awesome Mr Kalid . You really should have been my teacher :slight_smile:

You have no idea how happy you just made me.

You just made my brain do this.

@Anon1: Thank you!

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

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

One thing I still don’t understand is why S=O/H, C=A/H, T= O/A.


@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.

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?

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.

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.

@Anonymous: I put an answer at, 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 :).

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