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

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

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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