anonymous
  • anonymous
True or False? Can anyone explain these? Thank you The statement sin(sin^-1)=x for all real numbers in the interval -∞≤ X≤ ∞? The statement sin^-1(sinx)=x for all real numbers in the interval -∞≤ X≤ ∞?
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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freckles
  • freckles
First question doesn't make sense sin(sin^-1)=x? is that really what it says? For the second one ask yourself what is the domain of sin(x), then ask yourself what is the range of sin(x), then ask yourself is the range of sin(x) either a subset or equal to that of the domain of the sine inverse function.
anonymous
  • anonymous
Sorry, I left out an X. It says sin(sin^-1X)=x
freckles
  • freckles
what is the domain of the sine inverse function?

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anonymous
  • anonymous
sin domain all real numbers sin^-1 I'm not sure -1,1?
freckles
  • freckles
Right! \[y=\sin(x) \text{ has domain all real numbers } \text{ or you can say } \\ y=\sin(x) \text{ where} -\infty
freckles
  • freckles
now look at the first one the inside function is arcsin(x) which means we can only input what numbers?
anonymous
  • anonymous
0
freckles
  • freckles
you just said the domain was for arcsin(x) was [-1,1] why do you change it to just the number 0?
freckles
  • freckles
arcsin(x) has domain [-1,1] this means you can input any number in the interval [-1,1] and since the domain for sin(x) is all real numbers it doesn't matter what the output of arcsin(x) is... the domain of sin(arcsin(x)) will just be the domain of arcsin(x)
anonymous
  • anonymous
-1 ≤ 0 ≤ 1 Because 0 is greater than -1, but less than 1
freckles
  • freckles
you do know there are infinitely many numbers between -1 and 1?
freckles
  • freckles
0 is not the only number
anonymous
  • anonymous
Oh okay. so problem 1 is true and problem 2 will be false.
freckles
  • freckles
examples you have -1,-.99999,-.95,-.55, -.5, 0,.4, so on... you can not just write on the numbers in a listing type roster
freckles
  • freckles
No... the domain of sin(arcsin(x)) is the domain of arcsin(x)
freckles
  • freckles
you can see reason above
freckles
  • freckles
|dw:1446317266707:dw|
freckles
  • freckles
and so what is the domain of sin(arcsin(x))?
anonymous
  • anonymous
The domain of sin(arcsin(x)) is all real numbers between -1 and 1
freckles
  • freckles
right so sin(arcsin(x))=x when x is in [-1,1] choosing an x outside that interval for example x=2 would give us arcsin(2) which is undefined... so for the first one we only needed to consider the domain of arcsin(x)
freckles
  • freckles
so while arcsin(sin(x)) has domain all real number we will not have arcsin(sin(x))=x for all real numbers because arcsin doesn't have range all real numbers arcsin(sin(x))=x..... now this one sin(x) does have all real numbers but here you have to consider the domain restriction you put on y=sin(x) so that y=arcsin(x) could exist. what is that domain restriction?
freckles
  • freckles
I'm asking you what was the domain restriction put on y=sin(x) so y=arcsin(x) could exist? This question is also equivalent to what is the range of y=arcsin(x)?
anonymous
  • anonymous
-1 to 1
freckles
  • freckles
no that is the domain of y=arcsin(x) and the range of y=sin(x)
freckles
  • freckles
the restricted domain of y=sin(x) and the range of y=arcsin(x) is [-pi/2,pi/2]
anonymous
  • anonymous
Thank you
freckles
  • freckles
|dw:1446318461088:dw| some of the words got cut off
freckles
  • freckles
Examples: \[\arcsin(\sin(\frac{\pi}{4}))=\frac{\pi}{4} \\ \arcsin(\sin(\frac{ 5\pi}{4})) \neq \frac{ 5\pi}{4} \text{ since } \frac{5 \pi}{4} \text{ is not in the set of numbers } \\ \text{ that is consider the outputs (range ) of } \arcsin( )\] but... \[\arcsin(\sin(\frac{5\pi}{4})) \text{ does exist } \\ \sin(\frac{5\pi}{4})=\sin(-\frac{\pi}{4}) \text{ so } \arcsin(\sin(\frac{5\pi}{4}))=\frac{-\pi}{4}\] so I hope these examples made it more clear that arcsin(sin(x))=x only when x is in [-pi/2,pi/2] though arcsin(sin(x)) will also exist for other x, the output just won't be x
freckles
  • freckles
any questions on this question?

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