if sin alpha and cos alpha are the roots of equation ax2 + bx+c = 0, then b2 = ?

Question

actually the raise to the power can't be put i used simple

Answer 1

if sin alpha and cos alpha are the roots of equation ax2 + bx+c = 0, then b^2 = ?

if sin alpha and cos alpha are roots then x - sin(t) = 0,  x - cos(t) = 0. So we can recontruct the quadratic by multplying together the two factors of the quadratic, i.e.  (t = alpha)

(x - sin(t))(x - cos(t)) = 0

x^2 - x.cos(t) - x.sin(t) + sin(t).cos(t) = 0

x^2 - (sin(t)+cos(t)).x + (1/2)sin(2t) = 0

comparing this with the original quadratic,

b = -(sin(t) + cos(t))

b^2 = sin^2(t) + 2sin(t).cos(t) + cos^2(t)

b^2 = 1 + sin(2t)

and (1/2)sin(2t) = c, from the original quadrartic, so

b^2 = 1 + 2c