Discussion Overview
The discussion centers around solving the equation $$5\cos(4x)=4$$ for its third and fourth smallest solutions. Participants explore methods for finding solutions, the cyclical nature of the cosine function, and the implications of the equation's periodicity.
Discussion Character
- Homework-related
- Mathematical reasoning
- Conceptual clarification
Main Points Raised
- One participant describes their approach to finding the first two solutions using inverse cosine and division by 4, but expresses uncertainty about finding the third solution.
- Another participant suggests that there are infinitely many solutions and emphasizes the need to clarify which specific solutions are being sought.
- A later reply indicates that the instructor expects students to understand the cyclical nature of the cosine function and suggests adding multiples of $$2\pi$$ to the primary solutions to find additional solutions.
- One participant corrects a previous statement by noting that the period of the function is actually $$\frac{\pi}{2}$$ due to the factor of 4 inside the cosine function, rather than $$2\pi$$.
- Another participant acknowledges their oversight in not calculating the actual answers and recognizes the importance of the factor of 4 in the cosine function.
Areas of Agreement / Disagreement
Participants generally agree on the cyclical nature of the cosine function and the method of generating additional solutions, but there is some disagreement regarding the specifics of the periodicity and the exact solutions being sought.
Contextual Notes
There is a lack of clarity regarding the specific values of the first two solutions and how they relate to the third and fourth solutions. The discussion also reflects varying interpretations of the periodicity of the function based on the equation's structure.