Pure Mat Question ( 40 marks)

(a)
(cosA+isinA)^5=cos5A+isin5A
(cosA+isinA)^5
=cos^5A+5icos^4AsinA-10cos^3Asin^2A-10icos^2Asin^3A+5cosAsin^4A+isin^5A
compare the real part
cos5A
=cos^5A-10cos^3Asin^2A+5cosAsin^4A
=cosA(cos^4A-10cos^2Asin^2A+5sin^4A)
=cosA(cos^4A-10cos^2A(1-cos^2A)+5(1-cos^2A)^2)
=cosA(cos^4A-10cos^2A+10cos^4A+5(1-2cos^2A+cos^4A))
=cosA(cos^4A-10cos^2A+10cos^4A+5-10cos^2A+5cos^4A)
=cosA(16cos^4A-20cos^2A+5)
f(x)=16x^4-20x^2+5
f(cosA) = 0
from cos5A=0
5A=π/2, 3π/2, 5π/2, 7π/2, 9π/2
A=π/10, 3π/10, 5π/10, 7π/10, 9π/10
we know 5π/10 is for cosA=0
so the values of A (0 <= A <= pi) such that f(cosA) = 0 are
π/10, 3π/10, 7π/10, 9π/10
(b)
since π/10, 3π/10, 7π/10, 9π/10 makes f(cosA) = 0
so cosπ/10, cos3π/10, cos7π/10, cos9π/10 are the roots of f(x)
consider the coefficient of the highest degree ,we have
f(x)=16(x- cos(pi/10)) (x-cos(3pi/10) ) (x-cos(7pi/10) ) (x-cos(9pi/10) )
but f(x)=16x^4-20x^2+5
sub x=1
f(1)
=16
so
16(1- cos(pi/10)) (1-cos(3pi/10) ) (1-cos(7pi/10) ) (1-cos(9pi/10) ) = 1
(1- cos(pi/10)) (1-cos(3pi/10) ) (1-cos(7pi/10) ) (1-cos(9pi/10) ) = 1/16

because
(1-cos(7pi/10) )=1+cos( 3pi/10)
(1-cos(9pi/10) )=1+cos( pi/10)
(1- cos(pi/10)) (1-cos(3pi/10) ) (1-cos(7pi/10) ) (1-cos(9pi/10) ) = 1/16
(1- cos(pi/10)) (1-cos(3pi/10) ) (1+cos( 3pi/10) ) (1+cos( pi/10) ) = 1/16
(1- cos^2(pi/10))(1- cos^2(3pi/10))=1/16
sin(pi/10)sin(3pi/10 ) =1/4

Let z=cosA+isinA
cos5A
=(1/2)(z^5+z^-5)
=(1/2)[(cosA+isinA)^5+(cosA+isin(-A))^5]
=cos^5A - 10cos^3Asin^2A + 5cosAsin^4A
=cos^5A - 10cos^3A(1-cos^2A) +5cosA(1-cos^2A)^2

Hence
f(x)=x^4-10x^2(1-x^2)+5(1-x^2)^2=16x^4-20x^2+5

cos5A =cosA f(cosA)
if f(cosA)=0, cos5A=0
5A=pi/2, 3pi/2, 5p/2, 7pi/2, 9pi/2 (0&lt;=5A&lt;=5pi)
A=pi/10, 3pi/10, pi/2, 7pi/10, 9pi/10
But cos(pi/2)=0 also implies cos5A=0
and clearly x=0 is not a root of f(x)=0
Hence A=pi/10, 3pi/10, 7pi/10, 9pi/10

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cos(kpi/10) (k=1,3,7,9) are roots of 16x^4-20x^2+5=0
1-cos(kpi/10) are roots of 16(1-x)^4-20(1-x)^2+5=0
Product of roots=1/16
(1- cos(pi/10)) (1-cos(3pi/10)) (1-cos(7pi/10) ) (1-cos(9pi/10) ) = 1/16


cos(pi/10)=-cos(pi-pi/10)=-cos(9pi/10)
cos(3pi/10)=-cos(pi-3pi/10)=-cos(7pi/10)

(1- cos(pi/10)) (1-cos(3pi/10)) (1+cos(3pi/10) ) (1+cos(pi/10) ) = 1/16
(1- cos^2(pi/10)) (1-cos^2(3pi/10)) = 1/16
sin^2(pi/10)sin^2(3pi/10) =1/16
sin(pi/10)sin(3pi/10) =1/4