How can algebra and calculus be used to derive the identity for sin(a+b)?

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Discussion Overview

The discussion revolves around deriving the identity for sin(a+b) using algebra and calculus, with participants exploring various methods including geometric proofs, rotation matrices, and power series. The scope includes theoretical approaches and personal reflections on memorization versus derivation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant asks how to derive the identity for sin(a+b) using algebra or calculus, noting familiarity with geometric and imaginary number methods.
  • Another participant suggests using rotation matrices as a method for derivation, providing a link to further information.
  • A participant argues that pure algebra is insufficient for deriving the identity due to the transcendental nature of the sine function, suggesting that calculus, particularly through power series, offers a more rigorous approach.
  • One participant recalls a more intuitive geometric proof from high school and expresses a desire for a simpler derivation method for the sum and difference formulas, mentioning the usefulness of these formulas for deriving other trigonometric identities.
  • A participant references another thread that contains a specific proof, recommending it as charming and relevant to the discussion.
  • One participant expresses the view that the sine and cosine angle addition formulas are fundamental enough to warrant memorization rather than derivation.
  • Another participant agrees that one of the formulas can be proven easily if one is already given, specifically mentioning the sine formula.

Areas of Agreement / Disagreement

Participants express differing opinions on the necessity of memorizing the sine and cosine angle addition formulas versus deriving them. There is no consensus on a single preferred method for derivation, as various approaches are discussed.

Contextual Notes

Some participants highlight the limitations of algebraic methods for transcendental functions and the potential dependency on initial relations for algebraic derivations. The discussion reflects a range of personal experiences and preferences regarding memorization and derivation techniques.

FulhamFan3
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How do you find the identity for sin(a+b) using algebra or calculus? I already know how to do it with geometry and by using imaginary numbers.
 
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Pure algebra won't help -- sin is a so-called transcendental function. At least, algebra won't help unless you provide some initial algebraic relations to get started. (Though, this particular identity would usually be such a thing that you'd use to start, not something you'd derive)


As for calculus, of the most rigorous proofs, it is the easiest to follow. sin is simply a power series:

[tex] \sin z := \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n + 1}}{(2n+1)!}[/tex]

And similarly for cos. If you plug in z = p + q, then you can algebraically derive the identity from this.
 
I seem to remember a more intuitive "geometric" proof (if you could call it that) in high school. Just wondering, because that was back in the day before Maclaurin series, and it would be nice if there were an easier derivation (easier to remember) for the sum and difference formulas. The reason why I care is that I know how to derive the double angle formulas from the sum/difference ones in like two seconds, and I know how to get the half angle formulas from the double angle ones, so if I just knew how to get at the sum/difference ones quickly without memorizing them (a bit tedious, although they're sort of half-memorized already), then I'd have all these trig identities at my fingertips... I have a pretty good memory, but memorization is not the best way, IMO.

Edit: didn't read Fullham's post. he mentioned the geometric method. Gotta link? And how do you do it using imaginary numbers? The Euler identity?
 
Last edited:
There's a thread on this specific matter right in this subforum.It's called "2 proofs".You'll find the proof given by HallsofIvy really charming.

Daniel.
 
Frankly, I'd consider the sine and cosine angle addition formulas sufficiently fundamental that they're worth memorizing on their own merit, rather than trying to justify in terms of other concepts.
 
Well,only one of them is worth it.The other can be proven immediately,once one is "given"...

Let's chose the SINE... :wink:

Daniel.
 

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