Question

Find the implied domain and range of y=cos(2arcsinx)

Could someone please explain this to me step-by-step. Thanks

Answer 1

@rational I need your help

Answer 2

I know that 2arcsin x is restricted by cos

Answer 3

@sourwing

Answer 4

it could really help you if you knew how to write y=cos(2arcsin(x)) an algebraic expression.

Answer 5

do you know how to do that?

Answer 6

my keyboard doesnt allow me

Answer 7

when looking at 2arcsin x do i focus on its domain for step 1?

Answer 8

i know that the domain for arcsin (x) is [-1,1]

Answer 9

@freckles

Answer 10

Correct, so no matter what, you won't be able to input values outside of [-1,1], so that means [-1,1] is indeed your domain. How about your range, what kind of range does arcsin(x) give?

Answer 11

see that is something i dont know

Answer 12

do you mean if i plug in -1 and 1 into the equation?

Answer 13

Well, the range of arcsin(x) is [-pi/2, pi/2]. So that means that 2arcsin(x) will place values between -pi and pi into the cosine function. So if you take cosine from -pi to pi, what kind of final range will that give you?

Answer 14

where did you get -pi and pi from?

Answer 15

Because we have 2arcsinx. If arcsinx gives me -pi/2, it will get multiplied by 2 and become -pi. Same for if arcsinx gives us pi/2, it will be multiplied by 2 and give us pi.

Answer 16

ok

Answer 17

but it wont have any effect on the domain

Answer 18

?

Answer 19

No, it wont affect the domain. Thedomain is [-1,1]. But we will have values for cos(-pi) all the way through cos(pi) going around the unit circle. If we strictly look between -pi and pi, what will be the range of cosine?

Answer 20

so will the range represent the angle when we replace 2arcsin x with -pi and pi?

Answer 21

i get -1 for both

Answer 22

these are the same values i get if i plugged in -1 and 1 from the domain into the equation y=cos(2arcsin x)

Answer 23

i guess we have found the answer of the domain which is [-1,1]. Right?

Answer 24

i have posted this question a few times on this forum and it remained unanswered so i guess its not that easy to solve

Answer 25

Sorry for infrequent answers. But if you follow the values of cosine between -pi and pi, your cosine function will range from [-1,1]. Sure, -pi and pi give values of -1 for cosine, but 0 is in between -pi and pi, so you certainly will end up with cos(0) = 1 eventually.

The idea is because of the arcsinx, we can only input -1 to 1, hence domain [-1,1]. For the range, arcsinx will produce all values between -pi/2 and pi/2. Multiplied by the 2 on the outside, we end up having all values inside of cosine between -pi and pi. When following the graph of cosine between -pi and pi, your range ends up going from [-1,1]. So the domain AND range are [-1,1].