Critical Points of cos y & cos 2x - Why the Difference?

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SUMMARY

The critical points for the function cos y are defined as nπ - 0.5π, while for cos 2x, they are n/2 * π - 0.25π. The discrepancy arises when comparing algebraic and graphical methods; algebraically, the use of even integers for n constrains the results, whereas graphical analysis shows no such restriction. This indicates a potential misunderstanding between critical points and roots, suggesting a need for clarification in the definitions used.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with critical points and their significance in calculus
  • Knowledge of algebraic manipulation involving trigonometric identities
  • Basic graphing skills for trigonometric functions
NEXT STEPS
  • Study the definitions and differences between critical points and roots in calculus
  • Learn about the graphical representation of trigonometric functions
  • Explore the implications of using different values for n in trigonometric equations
  • Investigate the behavior of cos y and cos 2x through advanced graphing techniques
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and trigonometry, as well as anyone seeking to clarify the concepts of critical points and roots in trigonometric functions.

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for cos y the critical points are n*pi - 0.5*pi
for cos 2x, the critical points should be n/2 * pi - 0.25*pi
because if I set y=2x = n*pi - 0.5*pi , I get the eq in red for x.
However if I do it graphically, I just get n*pi - 0.25*pi.
The question: why algebraically I am constrained to use the even number n but graphically, there is no restriction on n?
 
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I think your critical points for cos y are wrong. Do you mean sin?

The question: why algebraically I am constrained to use the even number n but graphically, there is no restriction on n?
Answer: At least one of them is wrong.
 

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