In order for a trig function to have an asymptote, it must be undefined. We know that the ratio for sin is y/r and the ratio for cos is x/r. In a unit circle, the measure of r will always equal 1, thus, the value of sin and cos will never be undefined. As a result, there are no asymptotes. However the other trig functions do have asymptotes. Let's look at why this is. Cosecant has asymptotes where sine is equal to 0.This is because it is the reciprocal of sine, 1/sin. This means that when the value of y is 0, the value of cosecant is 1/0, making it undefined. Similarly, Cotangent has an asymptote when sin = 0 (y=0) because it's trig ratio is cos/sin. Because both cosecant and cotangents asymptote depend on when y = 0, they share the same asymptotes at (1,0) and (-1,0) which would be 0* and 180*, also referred to as 0 and pi in radians (the 2 points on the unit circle where the y values are 0). Secant is undefined when cos is equal to 0 (x=0). The value of secant would be 1/0 , making it undefined. Likewise, tangent is y/x , so whenever the x value= 0 , there is an asymptote. The x-value equals 0 at (0,1) and (0,-1) which can also be written as pi/2 and 3pi/2. Because of this, secant and tangent share the same asymptotes, both relying on their denominator of x.
Please refer to the picture below that identifies the different unit circle ratios.