a) Show sin is not a rational function. By definition of a rational function, a rational function cannot be 0 at infinite points unless it is 0 everywhere. Obviously, sin have infinite points that are 0 and infinite points that are not zero, thus not a rational function. b) Show that there do not exist rational functions f0,…,fn−1(x)(sin⁡x)n−1+…+f0(x)=0 for all x First choose x=2kπ, so f0(x)=0 for x=2kπ. Since f0 is rational ⇒f0(x)=0 for all x. Thus can write sin⁡x[(sin⁡x)n−1+fn−1(sin⁡x)n−2…+f1(x)]=0 Question: The second factor is 0 for all x≠2kπ How does this imply that it is 0 for all x? And how does this lead to the result?

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a) Show sin is not a rational function.    By definition of a rational function, a rational function cannot be 0 at infinite points unless it is 0 everywhere. Obviously, sin have infinite points that are 0 and infinite points that are not zero, thus not a rational function.    b) Show that there do not exist rational functions f0,…,fn−1(x)(sin⁡x)n−1+…+f0(x)=0 for all x    First choose x=2kπ, so f0(x)=0 for x=2kπ. Since f0 is rational ⇒f0(x)=0 for all x. Thus can write sin⁡x[(sin⁡x)n−1+fn−1(sin⁡x)n−2…+f1(x)]=0    Question: The second factor is 0 for all x≠2kπ How does this imply that it is 0 for all x? And how does this lead to the result?

nghycylluzn0 · Accepted Answer

Use continuity of sin⁡ and rational functions.

a) Show sin is not a rational function. By definition of a rational function, a rational funct

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