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Asymptotes of Secant, Cosecant, and Cotangent - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

To find the x-intercepts and asymptotes of secant, cosecant, and cotangent, rewrite them in terms of sine and cosine. Notice that since secant and cosecant have 1 in the numerator and a trig function in the denominator, they can never equal zero; they do not have x-intercepts. The vertical asymptotes of the three functions are whenever the denominators are zero.

I want to talk abut the asymptotes of the reciprocal trig functions secant, cosecant and cotangent recall the identities secant equals 1 over cosine, cosecant equals 1 over sine and cotangent equals cosine over sine these will help us identify the asymptotes. Let's start with secant, notice that secant is going to be undefined when cosine equals 0 and cosines equals 0 when theta equals for example pi over 2, 3 pi over 2, 5 pi over 2 and so on. But it's also 0 at negative pi over 2, negative 3 pi over 2 and so on.
How can we say this more briefly how can we say this more efficiently? Well we could say theta equals pi over 2 plus n pi plus any integer multiple of pi. This is pi over 2 plus pi, this is pi over 2 plus 2 pi, this is pi over 2 minus 1 pi etc. So that means that secant theta is going to have vertical asymptotes when theta equals pi over 2 plus n pi. So y equals secant theta, these would be the vertical asymptotes, now what about cosecant and cotangent, notice they both have sine and in the denominator and that means they're both going to have vertical asymptotes in exactly the same places when sine theta equals 0. And of course that happens when theta equals 0 pi, 2 pi et cetera but it also happens at negative pi, negative 2 pi and so on.
And these numbers are all just integer multiples of pi so I can express this more briefly as n pi where n is an integer, and this describes the vertical asymptotes of both y equals cosecant theta and y equals cotangent theta these are the vertical asymptotes. So very important you remember when especially when you're graphing but you might be asked a question about the asymptotes just by itself. The asymptotes of cosecant and cotangent are the integers multiples of pi, the asymptotes of secant are at pi over 2 plus the integers multiples of pi.

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