Why do the graphs of tan(x), sec(x), and csc(x) have asymptotes? How do we use the asymptotes to graph these trig functions?

The graphs of tangent, secant, and cosecant have vertical asymptotes because they are defined as ratios, and the denominator is occasionally zero. The asymptotes help to delineate sections of the graph.

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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...

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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.

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