Asymptoes of Rational Trigonometric Functions Example

Suppose we want to find the asymptotes of  
\[y=\frac{tanx}{sin(2x)+1}\]
.
We can some asymptotes by setting the denominator equal to zero.
\[sin(2x)+1=0 \rightarrow sin(2x)=-1 \rightarrow 2x=(2n+ \frac{3}{2}) \pi \rightarrow x=(n+ \frac{3}{4}) \pi\]

We cannot make  
\[x\]
  the subject, so cannot perform the same analysis to find asymptotes for  
\[y\]
. The function  
\[\frac{tanx}{sin(2x)+1}\]
  contains only trigonometric functions - no terms or factors such as  
\[e^x, \: x^3\]
  for example. This means that the function is periodic, in this case the function repeats every  
\[\pi\]
  radians. Because of this there can be no  
\[x\]
  asymptotes, or any other asymptotes of any kind. There only asymptotes are parallel to the  
\[y\]
  axis, given above.

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