Let y be the rational root of x^5 - 8x^4 + 2x^3 - 16x^2 + x - 8 = 0. Find sin ( π / y ).

Question

Let ABC be the triangle with AB = 1, AC = 3, and ∠BAC = π/2. If a circle of radius r > 0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is ___.

Answer 1

x^5 - 8x^4 + 2x^3 - 16x^2 + x - 8 = 0
x^5+2x^3+x -(8x^4+16x^2+8)=0
(x^2+1)^2*(x-8)=0
There is only one rational root --> x=8
So sin (π/8)= **0.************