Question

If sin and cos are the two roots to the equation 4x2-5x+a=0, then the value of a should be
answer= 9/8

Answer 1

replace  with θ for more traditional appearance

if sinθ and cosθ are roots of a quadratic, then the quad must be
k(x - sinθ)(x - cosθ) = 0
kx^2 - kxcosθ - kxsinθ + ksinθcosθ = 0
kx^2 - kx(cosθ + sinθ) + ksinθcosθ = 0

compare this with 4x^2 - 5x + a = 0 , then k = 4
and my equation becomes:

4x^2 -4x(cosθ + sinθ) + 4sinθcosθ = 0

further comparison:
-4x(cosθ+sinθ) = -5x
cosθ + sinθ = 5/4

and a = 4sinθcosθ

we know ....
(sinθ + cosθ)^2 = sin^2 θ + cos^2 θ + 2sinθcosθ
25/16 = 1 + (1/2)a
9/16 = a/2
a = 9/8

Yeahhh, you are right.

Answer 2

Thank you so much