question on arcsec infinity

Oct 2009
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can someone explain to me how arcsec infinity is equal to pi/2? thanks
 

HallsofIvy

MHF Helper
Apr 2005
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can someone explain to me how arcsec infinity is equal to pi/2? thanks
Strictly speaking, "\(\displaystyle arcsec(\infty)\)" isn't \(\displaystyle \pi/2\) or any other number. "\(\displaystyle \infty\)" isn't a number so secant is not defined for it. You could, of course, extend the real number system to include "infinities" but there are several different ways to do that- you would have to specify which you meant.

But, we can talk about this in the limit sense. \(\displaystyle sec(\theta)= \frac{1}{cos(\theta)}\) and as \(\displaystyle \theta\) goes to \(\displaystyle \pi/2\), \(\displaystyle cos(\theta)\) goes to 0 and so \(\displaystyle sec(\theta)= \frac{1}{cos(\theta)\) "goes to infinity" (get larger without bound). Thus, if we extend the real numbers sytem in that way, we have that "\(\displaystyle sec(\pi/2)= \infty\)" and so "\(\displaystyle arcsec(\infty)= \pi/2\)".
 
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