Confusion with product-to-sum trig identities

In summary, there are two identities listed on Wikipedia that involve sin and cos. They are very similar, but differ in the sign of one term. It can be confusing to know which one to use in a given situation, but they are essentially the same identity. Swapping the arguments and using the fact that sin of a negative angle is equal to the negative of sin of the positive angle, it is clear that they are equivalent.
  • #1
Jyan
36
2
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?
 
Mathematics news on Phys.org
  • #2
They're really the same identity - one of them being superfluous. If you swap the roles of the arguments, and use the fact that [itex]\sin(-x)=-\sin(x)[/itex], then you will see that they are the same.
 
Last edited:
  • #3
I see, thank you.
 
  • #4
You're welcome!
 
  • #5


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.
 

1. What are product-to-sum trig identities?

Product-to-sum trig identities are trigonometric identities that are used to rewrite expressions involving products of trigonometric functions into sums or differences of trigonometric functions. They are useful in simplifying and solving trigonometric equations.

2. How do I know when to use product-to-sum trig identities?

Product-to-sum trig identities are typically used when you have an expression that involves the product of two trigonometric functions, such as sin(x)cos(x). They can also be used when trying to simplify complicated expressions involving multiple trigonometric functions.

3. What are some common product-to-sum trig identities?

Some common product-to-sum trig identities include: sin(x)cos(y) = 1/2[sin(x+y) + sin(x-y)], cos(x)cos(y) = 1/2[cos(x+y) + cos(x-y)], and cos(x)sin(y) = 1/2[sin(x+y) - sin(x-y)].

4. How do I use product-to-sum trig identities to solve equations?

To use product-to-sum trig identities to solve equations, you will typically need to first rewrite the equation using the identities. This will allow you to simplify the equation and potentially factor or cancel terms to solve for the variable.

5. Are there any tips for remembering product-to-sum trig identities?

One tip for remembering product-to-sum trig identities is to think of them in terms of the sum and difference formulas for sine and cosine, which are commonly used to derive these identities. You can also practice using them in different types of problems to help solidify your understanding.

Similar threads

  • General Math
Replies
17
Views
4K
Replies
5
Views
1K
Replies
2
Views
928
Replies
4
Views
873
Replies
1
Views
669
Replies
9
Views
2K
Replies
4
Views
3K
Replies
4
Views
2K
  • Calculus
Replies
29
Views
514
Back
Top