Confusion with product-to-sum trig identities

Click For Summary

Discussion Overview

The discussion revolves around the confusion regarding the application of product-to-sum trigonometric identities, specifically the identities involving sinθcosβ and sinβcosθ. Participants explore the conditions under which each identity can be applied and the implications of swapping variables.

Discussion Character

  • Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses confusion about how to determine which product-to-sum identity to use when given sin x cos y, questioning if it matters which identity is applied.
  • Another participant argues that the two identities are essentially the same, suggesting that one is redundant and that swapping the arguments leads to the same result due to the property of sine.
  • A later reply acknowledges the clarification provided by the second participant.

Areas of Agreement / Disagreement

There is some disagreement regarding the necessity of both identities, with one participant asserting they are the same while another seeks clarity on their application. The discussion does not reach a consensus on the necessity of distinguishing between the two identities in practice.

Contextual Notes

The discussion highlights the potential for confusion in applying trigonometric identities and the importance of understanding the properties of sine when manipulating these identities. However, specific assumptions or definitions that might clarify the confusion are not fully explored.

Who May Find This Useful

This discussion may be useful for students or individuals studying trigonometry who are grappling with the application of product-to-sum identities and the nuances of trigonometric functions.

Jyan
Messages
36
Reaction score
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
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.
 
Last edited:
I see, thank you.
 
You're welcome!
 

Similar threads

High School Why do we care about trig identities?
  • · Replies 17 ·
Replies
17
Views
6K
Undergrad Adding trig functions with different amplitudes
  • · Replies 5 ·
Replies
5
Views
2K
High School Trigonometric Identity involving sin()+cos()
  • · Replies 11 ·
Replies
11
Views
2K
MHB Solving Trig Identity: Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Sum to Product Trigonometric identity does not work
  • · Replies 4 ·
Replies
4
Views
2K
MHB Trig Identity Problem: Solve cos2(x) + sin(x) = sin2(x) for 0^0<=x<=180^0
  • · Replies 9 ·
Replies
9
Views
3K
High School Confusion about the angle between two vectors in a cross product
  • · Replies 7 ·
Replies
7
Views
3K
High School Verifying trig identities.... what about when tan is undefined?
  • · Replies 2 ·
Replies
2
Views
2K
Using Euler's formula to prove trig identities using "sum to product" technique
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Confused about identity for product of cosines into a sum of cosines
  • · Replies 1 ·
Replies
1
Views
1K