We can define `tanx=(sinx)/(cosx)` . A fraction is undefined if the denominator is zero and the numerator is nonzero. Thus, tan(x) is undefined anywhere that cos(x)=0, or `{x|x=pi/2+n pi, n in ZZ}` . For each odd multiple of `pi/2` there is a vertical asymptote. As the period for tangent is...

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We can define `tanx=(sinx)/(cosx)` . A fraction is undefined if the denominator is zero and the numerator is nonzero. Thus, tan(x) is undefined anywhere that cos(x)=0, or `{x|x=pi/2+n pi, n in ZZ}` . For each odd multiple of `pi/2` there is a vertical asymptote. As the period for tangent is `pi` the graph repeats . The asymptotes act as a guide in you graph.

The secant is the reciprocal of the cosine and, like the tangent, has asymptotes at odd multiples of `pi/2` . However, the period is `2pi` , and the graph alternates between opening up and opening down.

The cosecant is the reciprocal of the sin function, so it has vertical asymptotes whenever sin(x)=0. `{x|x=n pi, n in ZZ}`

Like the secant the period is `2pi` and the function alternates opening up and down. The asymptotes help to guide as you graph.