Attachment 20678

the circle has radius 1.

OB=arcOA=$\displaystyle \theta$

Prove that OP= 1-cos$\displaystyle \theta$

Deduce that OD = $\displaystyle \frac{\theta(1-cos\theta)}{\theta-sin\theta}$

thanks.

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- Feb 3rd 2011, 11:03 AMBabyMiloGeometry Proof
Attachment 20678

the circle has radius 1.

OB=arcOA=$\displaystyle \theta$

Prove that OP= 1-cos$\displaystyle \theta$

Deduce that OD = $\displaystyle \frac{\theta(1-cos\theta)}{\theta-sin\theta}$

thanks. - Feb 3rd 2011, 11:55 AMemakarov
is it known that the circle's center lies on the line OP? Is C the center of the circle? If yes, it's pretty obvious that $\displaystyle CP = \cos\theta$.

For the second equality, let A' be the projection of A on OB. Use the fact that the triangle BA'A is similar to the triangle BOD. - Feb 3rd 2011, 12:07 PMBabyMilo
yes to your questions, but why cp=cos theta

i cant see it, im stupid :)

and can you explain the 2nd part in more details.

thanks. - Feb 3rd 2011, 12:26 PMemakarov
Because $\displaystyle \angle OCA=\theta$. The fact that $\displaystyle CP=\cos\theta$ follows from the definition of cosine.

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and can you explain the 2nd part in more details.