Simplification of Trigonometric Expression

Click For Summary
SUMMARY

The forum discussion centers on the simplification of the trigonometric expression $$\cos[k_n(L-\varepsilon)]-\cos k_nL$$. Participants clarify that for small epsilon, the correct simplification involves using the Taylor expansion, resulting in the approximation $$\sim k_n \varepsilon\, \sin(k_n L)$$ rather than the initially suggested $$2\sin k_nL$$. The cosine of a sum formula is also mentioned as a valid approach, but the Taylor expansion provides a clearer path to the correct result.

PREREQUISITES
  • Understanding of trigonometric identities, specifically the cosine of a sum formula.
  • Familiarity with Taylor series and Maclaurin expansion techniques.
  • Basic knowledge of limits and approximations in calculus.
  • Concept of small angle approximations in trigonometry.
NEXT STEPS
  • Study the Taylor series expansion for trigonometric functions.
  • Learn about the cosine of a sum formula and its applications.
  • Explore small angle approximations in trigonometry.
  • Practice simplifying trigonometric expressions using various identities.
USEFUL FOR

Mathematicians, physics students, and anyone involved in advanced calculus or trigonometry who seeks to deepen their understanding of trigonometric simplifications and approximations.

Dustinsfl
Messages
2,217
Reaction score
5
$$
\cos[k_n(L-\varepsilon)]-\cos k_nL
$$
I just read this for small epsilon this result can be further simpli ed by using the trigonometric identity for $\cos[k_n(L-\varepsilon)]$. Doing so result is
$$
2\sin k_nL
$$
I don't see this? Can someone explain?
 
Mathematics news on Phys.org
dwsmith said:
$$
\cos[k_n(L-\varepsilon)]-\cos k_nL
$$
I just read this for small epsilon this result can be further simplied by using the trigonometric identity for $\cos[k_n(L-\varepsilon)]$. Doing so result is
$$
2\sin k_nL
$$
I don't see this? Can someone explain?

It's not true, it is \(\sim k_n \varepsilon\, \sin(k_n L)\)

CB
 
CaptainBlack said:
It's not true, it is \(\sim k_n \varepsilon\, \sin(k_n L)\)

CB

What trig identity did you use though?
 
dwsmith said:
What trig identity did you use though?

You can use the cosine of a sum formula, but you get the same result by taking a Taylor expansion of the first term.

CB
 
Last edited:
CaptainBlack said:
You can use the cosine of a sum formula, but you get the same result by taking a Mclaurin expansion of the first term.

CB

If we use cosine sum, shouldn't it be
$$
\cos k_nL\underbrace{\cos k_n\varepsilon}_{\approx 1 \text{ for small }\varepsilon} + \sin k_nL\sin k_n\varepsilon - \cos k_nL = \sin k_nL\sin k_n\varepsilon
$$
 
Recall $\displaystyle \sin\theta\approx\theta$ for small $\displaystyle \theta$.
 

Similar threads

MHB 33. Express sin 4x in terms of sin x and cos x
  • · Replies 1 ·
Replies
1
Views
4K
MHB Is this Trigonometric Expression a Constant Function of x?
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Why is it important to convert angle units when using trigonometric functions?
  • · Replies 8 ·
Replies
8
Views
2K
MHB Write a trigonometric expression as an algebraic expression
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Determine ## \beta ## as a function of ##\theta## linkage
  • · Replies 2 ·
Replies
2
Views
2K
High School What is the use of simplifying a question in math?
  • · Replies 3 ·
Replies
3
Views
1K
MHB How to Solve the Trigonometric Equation 6*Sin^2(x) - 3*Sin^2(2x) + Cos^2(x) = 0?
  • · Replies 6 ·
Replies
6
Views
3K
High School Trigonometric substitution, a case I'd like to share
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Finding Minimal Mean Distance Curves on the Unit Sphere
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Tong Dynamics: cannot cancel angle from orbit energy expression calculation
  • · Replies 2 ·
Replies
2
Views
1K