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
thanks
Your solutions of a. and b. are correct. c. and d. however are wrong.
c. is probably obvious. As to d.: you must remember that $\displaystyle \cos^{-1}$ is only the exact inverse of $\displaystyle \cos x$ for $\displaystyle x\in [0;\pi]$. Since $\displaystyle \frac{5\pi}{3}\notin [0;\pi]$ we have to do some juggling to get the correct answer:
$\displaystyle \cos^{-1}\left(\cos\frac{5\pi}{3}\right)=\cos^{-1}\left(\cos \left(-\frac{5\pi}{3}\right)\right)$
$\displaystyle =\cos^{-1}\left(\cos\left(2\pi-\frac{5\pi}{3}\right)\right)=\cos^{-1}\left(\cos \frac{\pi}{3}\right)=\frac{\pi}{3}$
Ok, ok, but I take it that $\displaystyle \sin^{-1}(x)$ is just another way of writing $\displaystyle \arcsin(x)$, and if so then 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 $\displaystyle \sin^{-1}(c)$, which is either undefined or has a unique value; and asking for the solution(s) $\displaystyle x$ of the equation $\displaystyle \sin(x)=c$, which may have none or infinitely many.
Wikipedia???- bit of a dodgy source!
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.
.. and convention. If someone wants to go against the convention, he can do that - if he is prepared to bear the consequences: of being misunderstood all the time.
If a function, like $\displaystyle \arcsin$ 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 $\displaystyle \ln(e^5)$, I am sure to answer that this equals $\displaystyle 5$ without adding to that basic answer infinitely many different integral multiples of $\displaystyle 2\pi\cdot i$.
Not so, if someone asks me to determine the solution to an equation like $\displaystyle e^z=e^5$.
In the case of $\displaystyle \sin^{-1}$ this has the advantage of being a bit more specific as to what the solutions of $\displaystyle \sin(x)=c$ might be. They are $\displaystyle x=\sin^{-1}(c)+n\cdot 2\pi$, and $\displaystyle x=\pi-\sin^{-1}(c)+n\cdot 2\pi, n\in\mathbb{Z}$.
If, on the other hand, one takes $\displaystyle \sin^{-1}(c)$ to be the entire set of solutions, then all one can write is $\displaystyle x=\sin^{-1}(c)$. I really do consider this a disadvantage of that other convention (that I am not using, and that I hope few others are using).
Good points. It is so impotant then that teachers/lecturers/textbook writers etc all use the correct conventions even at a low level of instruction. As a teacher of senior students, one of my main issues is with having students "unlearn" misconceptions which have resulted from trying to keep instructions at a low level - eg when solving equations - "swap sides and swap signs" causes all sorts of issues later.
That's why I asked the origonal poster what the actual question said - seen many textbooks get it wrong.