Prove that, (cos π/7) *(cos 3π/7) *(cos 5π/7) are the roots of the equation 8x2-4x2-4x+1=0
Prove that, (cos π/7) *(cos 3π/7) *(cos 5π/7) are the roots of the equation 8x2-4x2-4x+1=0
Surya , 7 Years ago
Grade 11
Trigonometry> Prove that, (cos π/7) *(cos 3π/7) *(cos 5...
2 Answers
Arun
Dear Surya
As we see that
cos(π/7) + cos(3π/7) + cos(5π/7)
Multuiply and divide by 2sin(π/7)
= 1/2sin(π/7) [ 2sin(π/7)cos(π/7) + 2sin(π/7)cos(3π/7) + 2sin(π/7)cos(5π/7) ]
Use the following
sin(2A) = 2sinAcosA
2sinAcosB = sin(A+B) + sin(A-B)
= 1/2sin(π/7) [ sin(2π/7) + sin(π/7+3π/7) + sin(π/7-3π/7) + sin(π/7+5π/7) + sin(π/7-5π/7) ]
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) + sin(-2π/7) + sin(6π/7) + sin(-4π/7) ]
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) ]
= 1/2sin(π/7) [sin(6π/7)]
= 1/2sin(π/7) [sin(π - π/7)]
= [sin(π/7)]/2sin(π/7)
= ½
Multuiply and divide by 2sin(π/7)
= 1/2sin(π/7) [ 2sin(π/7)cos(π/7) + 2sin(π/7)cos(3π/7) + 2sin(π/7)cos(5π/7) ]
Use the following
sin(2A) = 2sinAcosA
2sinAcosB = sin(A+B) + sin(A-B)
= 1/2sin(π/7) [ sin(2π/7) + sin(π/7+3π/7) + sin(π/7-3π/7) + sin(π/7+5π/7) + sin(π/7-5π/7) ]
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) + sin(-2π/7) + sin(6π/7) + sin(-4π/7) ]
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) ]
= 1/2sin(π/7) [sin(6π/7)]
= 1/2sin(π/7) [sin(π - π/7)]
= [sin(π/7)]/2sin(π/7)
= ½
hence
these are the roots
Surya
It is not addition its multiplication
Prove that, (cos π/7) (cos 3π/7) (cos 5π/7) are the roots of the equation 8x2-4x2-4x+1=0
Pls solve this step by step.... Fast...
Pls solve this step by step.... Fast...
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