Is the Equation (1-x)Cosx = Sinx Continuous and Solvable in (0,1)?

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Discussion Overview

The discussion centers on the continuity and solvability of the equation (1-x)Cosx = Sinx within the interval (0,1). Participants explore the properties of the functions involved and seek to establish continuity using formal definitions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant requests help in proving that the equation is continuous.
  • Another participant asserts that since the functions 1-x, cos(x), and sin(x) are all continuous and no division is involved, continuity is evident.
  • A later reply acknowledges the continuity but seeks to prove it using the formal definition of continuity involving ε and δ.
  • Another participant questions whether the continuity of the individual terms can be assumed or if it needs to be proven first, suggesting that once established, the continuity of the whole expression follows since the absolute value of each term is bounded by 1.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of proving the continuity of the individual functions before concluding the continuity of the entire expression. There is no consensus on whether the continuity can be assumed or must be demonstrated.

Contextual Notes

Some participants highlight the need for formal definitions and proofs, indicating that assumptions about continuity may not be universally accepted without justification.

Who May Find This Useful

This discussion may be useful for students or individuals interested in mathematical analysis, particularly in understanding continuity and solving equations involving trigonometric functions.

ken1101
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prove that the equation (1-x)Cosx=Sinx has at least one solution in (0,1)

I am having some problem in proving that the equation is continous.

Please help. Thank you
 
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1-x , cosx, sinx are all continuous, and you are not doing any dividing, so continuity is obvious.
 
I realize it is continous. I just need prove that it is continuous using the definition of continuous

for every ε > 0 there exists a δ > 0 such that for all x ∈ I,: |x-c|<δ⇒|f(x)-f(c)|<ε
 
Can you assume each of the three terms are continuous or do need to first prove that? Once that is done, it is straightforward for whole expression since absolute value of each term is ≤ 1.
 

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