## 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.