Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Question

anonymous · Answer

cosine is always continous

mr.math · Answer

Yep!

anonymous · Answer

cosecant, secant and cotangent are discontinous at a point where their inverse are 0

mr.math · Answer

$\csc(x)=\frac{1}{\sin(x)}$, discontinuous at zeros of $\sin(x)$.

mr.math · Answer

Same thing with secant, discontinuous at zeros of $\cos(x)$.

diyadiya · Answer

how should i write it ?

mr.math · Answer

I will help you with co-secant:
Let $f(x)=\csc(x)=\frac{1}{\sin(x)}$. The zeros of $\sin(x)$ are $\frac{\pi}{2}+2n\pi$, where n is any integer. Thus f(x) is discontinuous at $\frac{\pi}{2}+2n\pi$, $n\in \Integer$.

anonymous · Answer

Cosine is continuous at all points.
Secant is discontinuous at the points where the value of the variant is odd multiple of pi/2
Cosecant is discontinuous at the points where the variant is a multiple of pi
Cotangent is discontinuous at the points where the variant is a multiple of pi.

mr.math · Answer

Crap! Zeros of $\sin(x)$ are multiples of $2\pi$.

mr.math · Answer

So cosecant is discontinuous at $x=2n\pi$, where n is any integer.

diyadiya · Answer

Yeah ! :)

mr.math · Answer

:D

mr.math · Answer

What I did above is the zeros of cos(x), which states the discontinuity of $\sec(x)$.

zarkon · Answer

isn't it just the multiples of $\pi$

zarkon · Answer

not $2\pi$

mr.math · Answer

Lol, what's wrong with me?!

diyadiya · Answer

is it?

mr.math · Answer

Yeah, as Zarkon said zeros of $\sin(x)$ are $n\pi$, and zeros of $\cos(x)$ are $\frac{\pi}{2}+n\pi$.

mr.math · Answer

Sorry for the confusion!

mr.math · Answer

And thanks for Zarkon!

diyadiya · Answer

And Tan ?

mr.math · Answer

What do you think?

diyadiya · Answer

hm lol idk :D

mr.math · Answer

Use the same concept (i.e find zeros of the denominator).

mr.math · Answer

$\tan(x)=\frac{\sin(x)}{\cos(x)}$, so?

mr.math · Answer

It would have the same discontinuity points as $\sec(x)$.

mr.math · Answer

While $\cot(x)$ will have the same discontinuity points as?

diyadiya · Answer

So cos x?

diyadiya · Answer

i mean discontinouos at zeroes of cos x?

mr.math · Answer

Yeah, tan(x) is discontinouos at zeros of $\cos(x)$, true!

mr.math · Answer

Try to reread the last few statements I made.

diyadiya · Answer

Sure :)

ThankYOu Libniz ,Mr.Math,Atchyut & Zarkon :)

mr.math · Answer

Good luck Diya! :D

diyadiya · Answer

Thanks :)