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Point \(P\) sits inside an angle \(O\) of \(60^\circ,\) as shown below.

 

Link to image --->    https://latex.artofproblemsolving.com/8/3/0/830265669c9c3e7ccc70574d2b1693251fc4e4c4.png

 

Line segments \(\overline{PA}\) and \(\overline{PB}\) are drawn so they are perpendicular to the two rays forming angle \(O,\) as shown. Given \(OA=a\) and \(OB=b\) find the distance \(OP\) in terms of \(a\) and \(b\). Explaining in depth would be greatly appreciated.

 

 

Thank you!

 Feb 4, 2020
 #1
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Let d = OP.  Then cos POA = a/d and cos POB = b/d, so POA = acos(a/d) and POB = acos(b/d).

 

You can then write cos(acos(a/d) + acos(b/d)) = cos(60) = 1/2.

 

Since cos(x + y) = cos(x) cos(y) - sin(x) sin(y), you can then plug into this formula to solve for d.

 Feb 4, 2020

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