General inverses of trigonometry function

[kntsy]

I find this statement confusing from Wikipedia:



$$sin(y)=x\Leftrightarrow y=arcsin(x)+2k\pi\ \forall\ k\in\mathbb Z$$


Is this statement false? "arcsin(x)" gives the principal value:[$\frac{-\pi}{2},\frac{\pi}{2}$]. Therefore,specifically, "arcsin(x)" gives $\frac{\pi}{3}\text{ but not}\frac{2\pi}{3}$.



My textbook:

$$sin(y)=x\Leftrightarrow y=(-1)^{k}arcsin(x)+k\pi\ \forall\ k\in\mathbb Z$$

I think my textbook's statement is more "complete".Or are the 2 statements both true?

 

[Mentallic]

Yes your textbook is giving all the solutions, while wikipedia is missing about half of the possible solutions. Take $$y=2\pi/3$$ for example, wikipedia won't return that solution with its formula while the other will.

 

[HallsofIvy]

I find this statement confusing from Wikipedia:



$$sin(y)=x\Leftrightarrow y=arcsin(x)+2k\pi\ \forall\ k\in\mathbb Z$$


Is this statement false? "arcsin(x)" gives the principal value:[$\frac{-\pi}{2},\frac{\pi}{2}$]. Therefore,specifically, "arcsin(x)" gives $\frac{\pi}{3}\text{ but not}\frac{2\pi}{3}$.



My textbook:

$$sin(y)=x\Leftrightarrow y=(-1)^{k}arcsin(x)+k\pi\ \forall\ k\in\mathbb Z$$

I think my textbook's statement is more "complete".Or are the 2 statements both true?

Look more closely at your textbook. Most texts use "Arcsin(x)" to mean the "principal value" of arcsin(x)- the value y such that sin(y)= x for $0\le y< \pi$.

Notice the difference? The CAPITALIZED A means the principal value, the small a arcsine, the more general value.

 

[Rasalhague]

Regarding capilatization conventions, Berkey/Blanchard: Calculus, 3rd ed. 1992, write, "Be careful to note that Tan^-1 x means the inverse of the function y = Tan x , not (tan x)^-1." Where Tan is the "restricted tangent function" with domain (-pi/2, pi/2) and range (-infinity, infinity). "The alternatives y = Arc tan x and y = arctan x are also frequently used to represent y = Tan^-1 x." Even so, they call Tan^-1, the "inverse tangent function", and Sin^-1 the "inverse sine function" (for -1 <or= x <or= 1 and -pi/2 <or=y <or= pi/2, the inverse sine function is defined by y = Sin^-1 x iff x = sin y). So for them, at least, it seems that arc- and Arc are interchangeable.

 

[CellCoree]

Both statements are true, but they are expressing different concepts. The first statement is stating that for any value of x, there are multiple possible values of y that satisfy the equation sin(y)=x. The principal value of arcsin(x) is just one of those possible values. The value of k determines which specific value of y is being referred to.

The second statement is expressing a more general concept, where the value of y can be any multiple of pi added to the principal value of arcsin(x). This allows for all possible solutions to the equation sin(y)=x to be included.

So, in summary, both statements are true and they are just expressing different aspects of the general inverse of the trigonometric function.