Confusion with product-to-sum trig identities

[Jyan]

I'm having some confusion with a couple trig identities. On wikipedia (http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities), the following two identities are listed:

sinθcosβ = (1/2)[sin(θ+β) + sin(θ-β)]

and

sinβcosθ = (1/2)[sin(θ+β) - sin(θ-β)]

I can see the difference between them if they are used with the same variables θ and β. But, how do you know which one is valid in any given situation? I find this a hard question to phrase, but I hope you can see my confusion. If you have sin x cos y, which identity can you apply? Does it matter? So long as you apply the other one to sin y cos x?

 

[Ackbach]

They're really the same identity - one of them being superfluous. If you swap the roles of the arguments, and use the fact that $\sin(-x)=-\sin(x)$, then you will see that they are the same.

 

[Jyan]

I see, thank you.

 

[Ackbach]

You're welcome!

 

[CellCoree]

I understand your confusion with these two identities. It is important to note that both identities are valid and can be applied in different situations. The key difference between them is the order in which the variables are used.

In the first identity, sinθcosβ, the sin function is applied to θ and the cos function is applied to β. This means that in any given situation, if you have the product of sin x and cos y, you can use this identity to rewrite it as (1/2)[sin(x+y) + sin(x-y)].

On the other hand, in the second identity, sinβcosθ, the sin function is applied to β and the cos function is applied to θ. This means that if you have the product of sin y and cos x, you can use this identity to rewrite it as (1/2)[sin(x+y) - sin(x-y)].

So, to answer your question, it does matter which identity you use depending on the order of the variables in the product. It is important to keep in mind that these identities are interchangeable and can be used in different situations.

In conclusion, when faced with a product of trigonometric functions, you can use either of these identities to rewrite it in terms of sums or differences. The key is to understand the order of the variables and apply the appropriate identity accordingly.