Evaluating cos((1/2)arccos(x))

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  • Thread starter Elissa89
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  • #1
Elissa89
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I don't know how to solve this, we didn't really cover any problems like this in class

cos(1/2*cos^-1*x)

This is due tonight online and would like help please.
 
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Answers and Replies

  • #2
MarkFL
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MHB
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Re: Need help, due tonight

The first thing I would consider is:

\(\displaystyle 0\le\arccos(x)\le\pi\)

Hence:

\(\displaystyle 0\le\frac{1}{2}\arccos(x)\le\frac{\pi}{2}\)

This means the cosine of the given angle will be non-negative. Next, consider the half-angle identity for cosine:

\(\displaystyle \cos^2\left(\frac{\theta}{2}\right)=\frac{1+\cos(\theta)}{2}\)

Given that the cosine function will be non-negative, we may write:

\(\displaystyle \cos\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+\cos(\theta)}{2}}\)

Can you proceed?
 

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