Are There Multiple Solutions for Sin(x) and Arcsin(x) at Pi/4?

[Karma]

For the problem sin-(1/sqrt2) ...(sin-.. being arcsin) the answer is pi/4 but is that the only answer because pi/4 lies between [-90,90]?


or would it also be right to say PI-4PI=3pi/4? (although i think this is wrong)

 

[HallsofIvy]

Do you know what the graph of sin(x) looks like? It goes up from (0,0) to ($\pi/2$, 1) then back down to ($\pi$,0). You are right that sin([itex]3\pi /4) is also equal to $\sqrt{2}{2}$ but IF YOUR PROBLEM SPECIFIES THAT THE ANSWER MUST BE BETWEEN $-\pi /2$ to $\pi/2$. If it does not then $3\pi /4$ is the "principal" value (it's the value your calculator gives you) so if you are asking that "tan^-1" be a single valued function, that would be its value. If you are solving "tan(x)= $\sqrt{2}/2$" then there are an infinite number of solutions- rhe two you give plus any multiple of 2 $\pi$.[/itex]

 

[Karma]

No the problem does not give the intervals of [-90,90]...just the question... so i can come to the conclusion that 3pi/4 is also correct (as is pi/4)

 

[Karma]

Halls.. My calculator gives me the answer of pi/4 though not 3pi/4 and where does the sqrt of 22 come from?

 

[HallsofIvy]

My "tex" messed up. It should have been $\sqrt{2}/2$.
Also the $3\pi /4$ was just a typo on my part. I meant, of course, $\pi/4$.
By the way- it is really bad practice to talk about "intervals [-90,90]" AND give values in terms of $\pi$. You are going to have to choose whether you are working in degrees or radians! (I strongly recommend radians.)

Again, if your problem is to find all solutions to $sin(x)= \sqrt{2}/2$, then the solutions are all numbers of the form $\pi/4 + 2n\pi$ and $3\pi/4+ 2n\pi$ where n is any integer. If your problem is to find $Sin^{-1}(\sqrt{2}/2)$ with arcsine as a single-valued function, then the only answer is $\pi/4$. (Notice the capital "S" on "Sin^-1". Many texts use the capital when they want to mean the single-valued function: the principal value.)