Deriving Every Trig Identity in a Minute

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[sudgylacmoe]

Read this before complaining that you need these identities to derive the power series: I'm using the power series as my definition of the sine and the cosine, not starting from the geometric definitions.

[angeldude]

I have trouble memorizing the trig identities anyways, and this video didn't really help with that, but it still was very cool to watch. I'll stick to rederiving any identities I need from the reverse Euler's Formulae and the power rule: exp(a+b) = exp(a)exp(b)

Reverse Euler's Formulae: cos(x) = (exp(ix) + exp(-ix))/2, sin(x) = (exp(ix) - exp(-ix))/2i. Yes, those two formulae are easier for me to remember than the trig identities in part because they're just special cases of ½(f(x) ± f(-x)) to split a function into even and odd parts.

[BlackEyedGhost]

Instead of defining them with the power series, you can define them in simpler terms using cis:

cis(θ) = i^(θ/90°)
cos(θ) = (cis(θ)+cis(-θ))/2
sin(θ) = (cis(θ)-cis(-θ))/2i

Then proving the identities is just a matter of applying power rules. You can also write the definitions above directly in terms of exponents using radians:

cos(θ) = (i^(2θ/π)+i^(-2θ/π))/2
sin(θ) = (i^(2θ/π)-i^(-2θ/π))/2i

[HoSza]

1. Missing at least 4 often used identities, those are in the form sin(a) + sin(b) = 2 sin((a+b)/2) cos((a-b)/2). (also 3 others: sin - sin, cos ± cos)

2. The half angle formula is incorrect because of the sign error on the intervals {[(2k-1)π, 2kπ] | k ∈ ℕ}.

[evandrofilipe]

Now I get a taste for the kind of trig I'll have to do next year (2024 - a year before 1st year undergrad). This looks painful to remember so hopefully I will find an easier way of deriving these.

[austinfogleman]

The derivative identities seem unsteady because you have to define the derivitives of sin and cos in order to create the taylor series unless there is a derivation of the taylor series without using derivitives which i am unaware of

[HoSza]

Sign correction for your half angle identity to allow it to work on all real numbers: sign(-x mod (2π) + π)

[leviathan]

Very nice! (btw, the github link in the description is kinda broken, youtube thinks the parentheses is part of the URL)

Also, what is Coq?

[caspermadlener]

The fact that sine and cosine have definitions in both algebra as well as geometry, which are equivalent on paper, but not in practice, is definely my favourite sesult in trigonometry.
I have even refered to this as the fundamental theorem of trigonometry!
Personally, I like the algebraic definitions as:
• sin(0)=0
• cos(0)=1
• sin'(x)=cos(x)
• cos'(x)=-sin(x)
I don't really like to define them using the Taylor series, since it adds infinitely terms together, which might (not in this case) converge.
Suppose that f"(x)=-f(x). Then, f(x)²+f'(x)² is constant, because its derivative is 2f(x)f'(x)–2f(x)f'(x)=0, so the function is constant.
For g(x)=f(x)–f(0)cos(x)–f'(0)sin(x), we still get g"(x)=-g(x), and g(x)²+g'(x)² is constant at g(0)²+g'(0)²=0²+0². Therefore, g(x) is always 0, so
f(x)=f(0)cos(x)+f'(0)sin(x).
Now, we get
sin(x+a)=sin(a)cos(x)+cos(a)sin(x),

This is an easier derivation, in my opinion, even though a minute long video is probably too short anyway.

[beinganangeltreon]

well, that is nifty, I did not know about negative trig identities are, trying to imagine polygons at the 4th quadrant (all negative) and how that differs from doing this guys math on 2nd quadrant (all positive) are they different? reply!

[hmyesmoment]

Chris Griffin if he was a math YouTuber

[insouciantFox]

Note: if you're using the half angle formulae and not the power reducing formulae, you're probably doing something wrong

[person]

how do you know f and g have to be those functions

[EricPham-grpg]

It it is fun for being in school then it OK but in reality it does not do what it suppose to do because time is spiral and as earth rotate the circle can not complete in space time electronic ect so need introduce and relate to times

[NXT_LVL_DVL]

You didn't derive them in a minute . You showed their deravtion within a minute .