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I'd like to know how to get the angle in the following problem:

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It is a square with side equal to 1. The radius of each semi circle is equal to the side of the square. How can this angle be determined?

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3 Answers 3

up vote 14 down vote accepted

It's $30^\circ$. Let the point of intersection of "upper" arcs $BD$ and $AC$ be called $E$, and of upper $BD$ with lower arc $AC$ be called $F$. You should recognize that $\triangle ABE$ is equilateral (why?). What about $\triangle ADF$? Now finish.

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Thank you, I understand now. –  mastergoo Apr 14 at 21:01

Drop a vertical line from the top intersection to the base of the square. You now have a right triangle whose hypotenuse is 1 and side adjacent is .5. .5 is the cos of the angle between upper line and base. Acos(.5) is 60 degrees.

In a similar fashion drop a vertical line from the lower interesection. Now you have aright triangle whose hypotenuse is 1 and the side opposite is point 5. .5 is the sin of the angle between hypotenuse and base. Asin(.5) is 30 degrees.

To get angle between these lines subtract 30 degrees from 60 degrees.

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Thank you, it is clean now... :) –  mastergoo Apr 14 at 21:00

This looks very similar to the construction of an angle of 60 degrees. You should be able to use that to find the angle.

http://www.mathopenref.com/constangle60.html

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