Find the coordinates of the vertices of a triangle inscribed in a unit circle given angles $ heta_1, heta_2, heta_3 $

Given the angles $ \theta_1, \theta_2, \theta_3 $ of the vertices of the triangle, the coordinates of the vertices on the unit circle are:

Vertex 1: $ ( \cos(\theta_1), \sin(\theta_1) ) $

Vertex 2: $ ( \cos(\theta_2), \sin(\theta_2) ) $

Vertex 3: $ ( \cos(\theta_3), \sin(\theta_3) ) $

Let

Given the angles $ heta_1, heta_2, heta_3 $, the vertices of the triangle are:

$ ( cos( heta_1), sin( heta_1) ) $

$ ( cos( heta_2), sin( heta_2) ) $

$ ( cos( heta_3), sin( heta_3) ) $

For $ heta_1 = frac{pi}{4} $, $ heta_2 = frac{pi}{2} $, and $ heta_3 = frac{3pi}{4} $:

$ ( cos(frac{pi}{4}), sin(frac{pi}{4}) ) = ( frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

$ ( cos(frac{pi}{2}), sin(frac{pi}{2}) ) = ( 0, 1 ) $

$ ( cos(frac{3pi}{4}), sin(frac{3pi}{4}) ) = ( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

The vertices of the triangle with angles $ heta_1, heta_2, heta_3 $ are:

$ ( cos( heta_1), sin( heta_1) ) $

$ ( cos( heta_2), sin( heta_2) ) $

$ ( cos( heta_3), sin( heta_3) ) $

For $ heta_1 = frac{pi}{4} $, $ heta_2 = frac{pi}{2} $, $ heta_3 = frac{3pi}{4} $:

$ ( frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

$ ( 0, 1 ) $

$ ( -frac{sqrt{2}}{2}, frac{sqrt{2}}{2} ) $

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