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A triangle ABC is circumscribed by a circle with center O. The arcs AB , BC and AC subtend the angles alpha , beta and gamma respectively at O.

 

If the area of the circle is 50π square units and that of the triangle is 15 square units , find the value of sin alpha + sin beta + sin gamma.

 May 22, 2020
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Since the area of the circle is 50·pi, we can find the radius of the circle:

  A  =  pi·r2     --->     50·pi  =  pi·r2     --->     r2  =  50     --->     r  =  sqrt(50)

 

The area of the triangle is 15.

This triangle can be subdivided into three triangles OAB, OBC, and OCA.

 

The area of each of these triangles can be found by using this formula:  A  =  ½·a·b·sin(C)

 

For each triangle, a = b = sqrt(50), so the area of the three triangles added together is:

  A  =  ½·sqrt(50)·sqrt(50)·sin(alpha) + ½·sqrt(50)·sqrt(50)·sin(beta) + ½·sqrt(50)·sqrt(50)·sin(gamma)

 

Factoring  A  =  ½·sqrt(50)·sqrt(50)·[ sin(alpha) + sin(beta) + sin(theta) ]

 

Since this area is 15:

  15  =  ½·sqrt(50)·sqrt(50)·[ sin(alpha) + sin(beta) + sin(theta) ]

  15  =  25·[ sin(alpha) + sin(beta) + sin(theta) ]

  15/25  =  [ sin(alpha) + sin(beta) + sin(theta) ]

  0.600  =  sin(alpha) + sin(beta) + sin(theta)

 May 22, 2020

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