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Question on complex numbers..Q. Let Po,P1,P2,P3,P4,.........,P(n-1) are vertices of a regular polygon in a unit circle,then prove that-(a)PoP1*PoP2*PoP3*..............PoP(n-1)=n(b)sin pi/n * sin 2pi/n * sin 3pi/n*........*sin[(n-1)pi/n]=2/2^(n-1).

Amit Saxena · Accepted Answer

we can see these pt. P0,P1,P2,P3...Pn-1 as the roots of unity as lie on a unit circle

and vertices of a REGULAR polygon in the argand plane the distance of all these pts. will be same and also they are seperated by equal angles 2pie/n

so it has n roots e^(ik2pie /n) k = 0 .........n-1

(z-z1)....(z-zn-1) = z^n-1 +z^n-2 .........+1

now take z = 1,which is P0 , we getPoP1*PoP2*PoP3*..............PoP(n-1)=n
where (z-z1) = P1P0 and so on

origin is centre of the regular polygon. and line joining origin and P0 is the real axis

b) (1-z1)(1-z2)....(1-zn-1) = n

mod (1-zk) = mod {1-cosk(2pie/n) - sin(2pie/n)k}

mod(1-zk) = 2 sin (piek /18)

multiplying all these values we get from k = 0 to n-1

sin pi/n * sin 2pi/n * sin3pi/n*........*sin[(n-1)pi/n]=2/2^(n-1)

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Question on complex numbers..
Q. Let Po,P1,P2,P3,P4,.........,P(n-1) are vertices of a regular polygon in a unit circle,then prove that-
(a)PoP1PoP2PoP3..............PoP(n-1)=n
(b)sin pi/n sin 2pi/n * sin 3pi/n........sin[(n-1)pi/n]=2/2^(n-1).

Amit Saxena , 12 Years ago

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Question on complex numbers..Q. Let Po,P1,P2,P3,P4,.........,P(n-1) are vertices of a regular polygon in a unit circle,then prove that-(a)PoP1*PoP2*PoP3*..............PoP(n-1)=n(b)sin pi/n * sin 2pi/n * sin 3pi/n*........*sin[(n-1)pi/n]=2/2^(n-1).

Amit Saxena , 12 Years ago

Amit Saxena

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Please send the full solution as soon as possible 🙏

discuss with askiitians tutors

I got 80% in jee mains and am from sc category . Will i get a good NIT

discuss with askiitians tutors

Looking for an integrated JEE+NEET Coaching Classess for 2 years along with an ongoing normal syllabus related classes followed by regular revision tests, please.ThanksGopinath [F/O Class 10th Student-Qatar]

discuss with askiitians tutors

Ntse coaching for class 9 ncert student, fee and schedule

discuss with askiitians tutors

Can a class 9 student take admission in the course, 2- what is the duration and mode of full course ie how many student in one class.

discuss with askiitians tutors

Question on complex numbers..
Q. Let Po,P1,P2,P3,P4,.........,P(n-1) are vertices of a regular polygon in a unit circle,then prove that-
(a)PoP1PoP2PoP3..............PoP(n-1)=n
(b)sin pi/n sin 2pi/n * sin 3pi/n........sin[(n-1)pi/n]=2/2^(n-1).

Looking for an integrated JEE+NEET Coaching Classess for 2 years along with an ongoing normal syllabus related classes followed by regular revision tests, please.
Thanks
Gopinath [F/O Class 10^th Student-Qatar]