The concern is provided in the chap. 1.4 that the publication titled: Calculus troubles for the new century, by Robert Fraga.

You are watching: Sin(arcsec(x))

What is the domain the $operatornamearcsec(sin x)$?

I feel puzzled as $operatornamearcsec( x ) = cos( x )$, as $sec( x) =arccos( x )$. I hope the am not be make error in regards to domain values here, as domain of $cos( x) $ is entire real number line; while that of $sec( x )$ is in the term $-1,cdots 1$.

So, the concern reduces to finding domain the $cos(sin x)$.

The domain of $cos(x)$ is the selection of $sin(x)$, therefore domain of input to $cos(x)$, is $-1le xle 1$.

But, the book states answer as below, in an unanticipated way:

The $operatornamearcsec(x)$ is identified only ~ above $\,$. However, $|sin, x,|ge 1$ just if $x$ is an odd lot of of $fracpi2$, therefore the domain consists of strange multiples of $fracpi2$.


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$$operatornamearcsec(x) extor sec^-1(x) e cos(x)$$

That declare is wrong. When we create something prefer $sec^-1x$, the does not typical $frac1secx$. It way "inverse secant function". It"s simply a notation thing. I know it"s confusion, but that"s just exactly how things room in modern-day inter-base.netematics.

As for the concern itself, the range of the role $sinx$ is $<-1,1>$. The train station secant function, $operatornamearcsec(x)$ (many times created as $sec^-1x$), is only identified on $(-infty,-1>cup<1,+infty)$. The intersection the those two sets is composed of the following two elements: $-1,1$. So, the domain that $f(x)=sec^-1(sinx)$ have to be all worths of $x$ whereby $sinx$ equals either $-1$ or $1$:

$$igg\fracpi2+kpi, kininter-base.netbbZigg.$$


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Michael RybkinMichael Rybkin
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The arcsec is no the mutual of the $sec$ function. Examine the an interpretation of arcsec.


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Steve KassSteve Kass
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You mix up the inverse and also the reciprocal function: $sec(x) = frac1cos(x) e arccos(x)$.


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