Given a triangle with sides 4, 5 and 8, what are the angles of the triangle?  

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We have the length of the sides of the triangle given as 4 ,5  and 8. Let us denote the sides as a = 4, b = 5 and c= 8. The angles A, B and C are opposite the three sides.

The angles of the triangle can be determined using the law of cosines.

a^2 = b^2 + c^2 – 2*b*c *cos A

=> cos A = [5^2 + 8^2 – 4^2]/ 2*5*8

=> cos A = ( 25 + 64 – 16)/80

=> cos A = 73/ 80

=> A = arc cos 73/ 80 = 24.14

Similarly cos B = (a^2 + c^2 – b^2)/ 2*a*c

=> cos B = [ 4^2 + 8^2 – 5^2]/2*4*8

=> cos B = 55/64

=> B = arc cos ( 55/64) = 30.75

And cos C = ( 4^2 + 5^2 – 8^2) / 2*4*5

=> cos C = -23/ 40

=> C = arc cos (-23/40) = 125.09

The angles of the triangle are 24.14 degrees, 30.75 degrees and 125.09 degrees.

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