Prasun Biswas' Messageboard

1) Because it doesn't.

2) Achieving a Fields medal someday. $[$Probability of this event $=P=\displaystyle \lim_{x\to \infty} \dfrac{1}{x}]$

3) Because I love pissing people off. In fact, that's what I do most of the time on Facebook or 9GAG or Omegle :3

4) Future? What's that? A new Pokemon that I haven't heard of?

5) Yeah... I can eat almost anything that's edible, given that it's cooked properly.

Answering that second question of yours took a lot of time. The rest took only a few minutes. xD

Well, I do math just because I love doing it. There's no other reason whatsoever!

S.S.T? Do you mean "Social Studies"?

I'll try to help as much as I can. But I may not be much active here for a while because my board exams are nearing.

Could you help me fill up my contest with many participants.

Haha, don't bother. I don't really care about my messageboard. I made it just because everybody else were making one for themselves. I spend most of my time in Facebook and so I mostly chat with my friends there. I rarely check this messageboard anyway.

First of all, yes, I'm 16. To be precise, I'm 16 years 10 months old as of today and am studying in class 12 of the academic year 2014-15. Just for the record, my board exams are going on. That's the reason that I'm less active for the time being and am replying to your comment now. Sorry for the late reply though. :)

Second of all, are you being sarcastic? Because I don't think I have any knowledge whatsoever. I just implement methods which I learnt by scouring the Internet for resources, etc in solving problems. Almost anyone who puts even a bit of self-effort into learning stuff can do that.

And finally, which solutions of mine are you unable to understand? If you can't understand a part, I expect you to make a comment there. The members of this community (including me) are always there to help others. :)

Its very easy. I will tell you it later on. Busy right now.

Are the cards drawn with replacement, i.e., are the cards refilled after each draw?

First of all, sorry for replying late. The thing is, my internet connection is screwed up for now and hence I'll be less active than usual for the next few days. I'm currently in a cyber cafe.

Now, coming onto your question, why would anyone memorize the values of $\arcsin(\sin x)$ ? There is a nice way to calculate the values. You simply need to understand 2 things first: (1) the function $g(x)=\arcsin(x)$ is the inverse function of $f(x)=\sin(x)$ and, (2) Since invertible functions are bijections, the domain and range are set in such a way that the domains and ranges are continuous as well as they form bijections.

Following identities will help you:

$\bullet\quad\arcsin(\sin x)=x~,~\textrm{iff } x\in \left[-\frac{\pi}{2},\frac{\pi}{2}\right]\\ \bullet\quad\sin(\pi-x)=\sin x\\ \bullet\quad\sin(-x)=-\sin(x)$

There is a trick to remembering some of these identities. Just remember the following form ($x$ is acute and $n$ be a positive integer):

$\sin\left(n\cdot\frac{\pi}{2}\pm x\right)=\begin{cases}\sin x~,~\textrm{if } n\textrm{ is even and the total angle in the parameter falls on 1st or 2nd quadrants}\\ -\sin x~,~\textrm{if } n\textrm{ is even and the total angle in the parameter falls on 3rd or 4th quadrants}\\ \cos x~,~\textrm{if } n\textrm{ is odd and the total angle in the parameter falls on 1st or 2nd quadrants}\\ -\cos x~,~\textrm{if } n\textrm{ is odd and the total angle in the parameter falls on 3rd or 4th quadrants}\end{cases}$

A question naturally comes to find. Why $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ ? Why not any other interval? The answer is simple. We need a continuous domain for $f(x)=\sin x$ or (say) a continuous range for $f(x)=\arcsin(x)$ that can produce all possible values that repeat over the entire reals. Simply observing the graph of $f(x)=\sin x$ shows that the selected interval is the best. Some may say, why not $\left[\frac{\pi}{2},\frac{3\pi}{2}\right]$ ? It produces the same result too. The answer for that is also simple and straightforward. We try to find ranges for trigonometric functions within the principal range $[-\pi,\pi]$ so as to keep it simple.

If you say, I can provide some worked out examples later, but I won't be much active for the next few days till my internet connection is up and running again.

Is this some kind of joke ?

Nope. I heard about RMO the first time just a year ago when I was already in class 12. It was until 2013 that even class 12 students were eligible to attempt for RMO but that changed in 2014 when class 12 students were no longer eligible to attempt for RMO. As a result, I never got the chance to attempt for RMO, let alone qualify for it.

In any case, I'm not good with olympiads and stuff. I'm not good with exams. Time limits scare me. :|

It's a good thing I never did attempt for RMO. I probably wouldn't have qualified for it anyway.

Swapnil I think you have great interest in Physics right ?