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natnatwebb
Without using a calculator, find the value for f(pi/3) if the function f(x)=sin^-1(Cos x). Give exact answers. The dependent variable is in radians.
\[f(\frac{ \pi }{ 3}) \] if the function \[f(x)=\sin^{-1} (\cos x)\]
so our equation would update to \[f (\frac{ \pi }{ 3})=\sin^{-1} (\cos \frac{ \pi }{ 3 })\] I think.
If you don't know the value of cos(pi/3) by heart (shame on you!), you can find it with the unit circle...pi/3 equals 60 degrees, so if you mark 60 deg on the unit circle, you will get the well-known(?) 30-60-90 triangle with the sides we all know and love: ½, 1 and ½√3. One of these is cos(pi/3)...
Okay, so then we have three possible equations, what do we do then?
Look at this image and remember where you can find cos 60 degrees in it:
I understand that the cos(pi/3)=60 degrees, but I'm not even sure I understand what I'm trying to find... I'm sorry I'm really not trying to be thick.
You need to find \[\sin^{-1}(\cos(\frac{ \pi }{ 3 }))\]so I would first try to figure out what cos(pi/3) is. (work from the inside to the outside, so to speak...) Now cos(pi/3) = cos(60 degrees) = ½. You know that, I think. Now you are left with :\[\sin^{-1}(\frac{ 1 }{ 2 })\]Doesn't that sound familiar? You are looking for an angle of which the sine is ½. That is also a well known angle...