- #1
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.
How can algebra and calculus be used to derive the identity for sin(a+b)?
- Thread starter FulhamFan3
- Start date
In summary, the conversation discussed different methods for finding the identity for sin(a+b). Algebra and calculus were mentioned as possible approaches, but it was noted that pure algebra would not be helpful due to sin being a transcendental function. The use of rotation matrices and power series was suggested as a way to derive the identity. The conversation also touched on the usefulness of memorizing the angle addition formulas and the possibility of deriving them from other concepts.
- #2
Ryoukomaru
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Here is how to do it by using rotation matrices:
https://www.physicsforums.com/showthread.php?t=61109&page=2
(scroll down a bit)
https://www.physicsforums.com/showthread.php?t=61109&page=2
(scroll down a bit)
- #3
Hurkyl
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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.
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.
- #4
cepheid
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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?
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:
- #5
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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.
Daniel.
- #6
Hurkyl
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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.
- #7
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Well,only one of them is worth it.The other can be proven immediately,once one is "given"...
Let's chose the SINE...
Daniel.
Let's chose the SINE...
Daniel.
FAQ: How can algebra and calculus be used to derive the identity for sin(a+b)?
What is the purpose of deriving trig identities?
Deriving trig identities is a technique used to simplify and manipulate trigonometric expressions. It allows us to express complex trigonometric functions in terms of simpler ones, making them easier to solve and work with.
How do you derive a trig identity?
To derive a trig identity, we use algebraic manipulations and trigonometric identities to rewrite the expression in a different form. This involves using properties such as the Pythagorean identities, sum and difference identities, and double angle identities.
What are the most commonly used trig identities?
The most commonly used trig identities include the Pythagorean identities (sin^2x + cos^2x = 1), the sum and difference identities (sin(x+y) = sinxcosy + cosxsiny), and the double angle identities (sin2x = 2sinxcosx).
Why is it important to know how to derive trig identities?
Knowing how to derive trig identities is important in many areas of mathematics, physics, and engineering. It allows us to simplify and solve complex trigonometric equations, and also helps us understand the relationships between different trigonometric functions.
Are there any tips for effectively deriving trig identities?
Yes, some tips for effectively deriving trig identities include: familiarizing yourself with the basic trigonometric identities, using algebraic manipulations to rewrite expressions, and looking for patterns and relationships between different trigonometric functions.