Half angle identity

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kaebun

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Sep 11, 2005
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use a half- angle identity to find the exact value of cos(67.5)
i have no clue what the half angle identity is or how to use it :?
 
Half-angle identities:

\(\displaystyle \frac{1-cos(x)}{2}=sin^{2}(\frac{x}{2})\)

\(\displaystyle \frac{1+cos(x)}{2}=cos^{2}(\frac{x}{2})\)

\(\displaystyle \frac{1-cos(2x)}{1+cos(2u)}=tan^{2}(x)\)
 
Do you know (or can reason from cos(135)=sin(90-135)=sin(-45)=-sin(45) ) the exact value of cos(135), Kaebun?

(Just wondering, are you familiar with "double" angle identities?)
 
yes ive done that before im not sure thats what we called it though
 
Correct.

From Galactus's second identity, you have cos(x) = cos(135) and can find cos(x/2), ie. cos(67.5).
 
oh so 135 is two time 67.5 so the answer would be the (sqr2/2)/2 ?
 
No. Use the identity.

\(\displaystyle \mbox{ \cos{(67.5)} = \cos{\left(\frac{135}{2}\right)} = \sqrt{\frac{1}{2}\left(1 + \cos{(135)}\right)} = \sqrt{\frac{1}{2}\left(1 - \frac{\sqrt{2}}{2}\right)}}\)

Test your radical simplifying skills.
 
those skils are nonexistant so i did sqr1/2 (sqr(1-(srq2)/2))i don't know what else to do
 
Not bad. Practise makes existent.

How about: \(\displaystyle \mbox{ \sqrt{\frac{1}{2}\left(1 - \frac{\sqrt{2}}{2}\right)} = \sqrt{\frac{1}{2}\left(\frac{2 - \sqrt{2}}{2}\right)} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}}\)
 
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