Question

if sin theta and cos theta are roots of ax^2-bx+c then what is relation between a,b,and c

Answer 1

let theta =@
according to question sin@ and cos@ are the roots of ax^2 - bx + c =0
we know ,
sum of roots = - coefficient of x /coefficient of x^2
sin@ + cos@ = b/a ----------(1)

product of roots = constant /coefficient of x^2
sin@ . cos@ = c/a ----------(2)

equation (1) take square
(sin@ +cos@ )^2 = b^2/a^2
sin^2@ + cos^2@ +2sin.cos@ =b^2/a^2
1 + [email protected]@ =b^2/a^2

put equation (2) value
1 + 2c/a = b^2/a^2
a^2 +2ac =b^2

Answer 2

Let two roots of the equation are p and q
Given
p=sinθ,q=cosθ
Sum of the roots =b/a
Sinθ+cosθ=b/a---(1)
Product of the roots=c/a
Sinθcosθ=c/a---(2)
Do the square of (1)
(sinθ+cosθ)^2=(b/a)^2
Sin^2θ+cos^2θ+2sinθcosθ=b^2/a^2
1+2c/a=b^2/a^2
(a+2c)/a=b^2/a^2
a+2c=b^2/a
Multiply each term with a
a^2+2ac=b^2