Why no asymptotes for sine/cosine?
The trig functions that we have come to know off the unit circle can also be expressed through ratio identities. These identities allow us to make sense out of multiple trigonometric functions. We know that when a ratio has a denominator of zero, that value is undefined. This undefined results in asymptotes when graphing trig functions.
The ratios for sine and cosine are opposite/hypotenuse (y/r) and adjacent/hypotenuse (x/r) respectively. If we were trying to find asymptotes for those two graphs, we might try to find an instance where r equals zero. However, r is not zero when observing a unit circle because the radius is always one.
If r can not be zero, then sine and cosine can never be undefined. Because they can not be undefined, sine and cosine do not have asymptotes.
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