Can someone check these for me?
a. sin(sin^-1(1/2)) = 1/2
b. cos(cos^-1(-1)) = -1
c. sin^-1(sin(pi/4)) = 3pi/4
d. cos^-1(cos(5pi/3) = 5pi/3
Inverse trigonometric functions - Wikipedia, the free encyclopedia will give you the rest of the story. A story that fits nicely with what I argued. So in this particular case, your average pocket calculator is absolutely right.
Moral: We must clearly distinguish between asking for the value of the term , which is either undefined or has a unique value; and asking for the solution(s) of the equation , which may have none or infinitely many.
I've always understood that y = arcsin x or y = sin ^(-1) x is NOT a function. Think of the graph y=sin x reflected in y=x.
I have seen it written Sin(-1) x with an upper case S to indicate a function and therefore the principle argument.
All comes down to definition I suppose.
If a function, like can be said to have "multiple branches", by convention one sticks to the "principal branch".
Same game with the natural logarithm: There are infinitely many branches of the "inverse of the natural exponential function" as well, but if someone writes , I am sure to answer that this equals without adding to that basic answer infinitely many different integral multiples of .
Not so, if someone asks me to determine the solution to an equation like .
In the case of this has the advantage of being a bit more specific as to what the solutions of might be. They are , and .
If, on the other hand, one takes to be the entire set of solutions, then all one can write is . I really do consider this a disadvantage of that other convention (that I am not using, and that I hope few others are using).
That's why I asked the origonal poster what the actual question said - seen many textbooks get it wrong.