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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: zool3 on October 22, 2011, 11:01:37 pm

Title: Need help with a question
Post by: zool3 on October 22, 2011, 11:01:37 pm: Can anyone do a step by step explanation of how to get from cos (arcsin x) to √(1- x^2)

Thanks :)
Title: Re: Need help with a question
Post by: HarveyD on October 22, 2011, 11:12:59 pm: let u = arcsin x and use chain rule
i.e
dy/dx = dy/du x du/dx

Edit: assumed you were looking for the derivative...
whoops
Title: Re: Need help with a question
Post by: Special At Specialist on October 22, 2011, 11:18:27 pm: cos(arcsin(x))
Let u = arcsin(x)
sin(u)
= sin(arcsin(x))
= x
We know that sin^2(x) + cos^2(x) = 1, so:
cos^2(x) = 1 - sin^2(x)
cos(x) = sqrt(1 - sin^2(x))
We worked out that sin(u) = x, so:
cos(u) = sqrt(1 - x^2)
cos(arcsin(x)) = sqrt(1 - x^2)
Title: Re: Need help with a question
Post by: kamil9876 on October 22, 2011, 11:20:28 pm: Poster didn't mention anything about derivative, I'm assuming it's to actually show they are equal. Hint: use $cos^2x+sin^2x=1$ . Also you should justify why you take the positive square root.
Title: Re: Need help with a question
Post by: zool3 on October 22, 2011, 11:30:38 pm: Quote from: Special At Specialist on October 22, 2011, 11:18:27 pm
cos(arcsin(x))
Let u = arcsin(x)
sin(u)
= sin(arcsin(x))
= x

why did you do this step?
Title: Re: Need help with a question
Post by: Special At Specialist on October 22, 2011, 11:42:08 pm: Quote from: zool3 on October 22, 2011, 11:30:38 pm
Quote from: Special At Specialist on October 22, 2011, 11:18:27 pm
cos(arcsin(x))
Let u = arcsin(x)
sin(u)
= sin(arcsin(x))
= x

why did you do this step?

To put it in the form:
sin^2(u) + cos^2(u) = 1
Since I knew what sin(arcsin(x)) was, it was only a matter of transposition to solve for cos(arcsin(x)).
Title: Re: Need help with a question
Post by: zool3 on October 22, 2011, 11:56:52 pm: ohh okay! thanks for that! :)