Prove that sin &#x2061;<!-- ⁡ --> &#x03C0;<!-- π --> 7 </mfrac> sin &#

Jorge Lawson

Jorge Lawson

Answered question

2022-05-22

Prove that sin π 7 sin 2 π 7 sin 3 π 7 = 7 8

Answer & Explanation

ryancameron52

ryancameron52

Beginner2022-05-23Added 8 answers

Since sin k π 7 = sin ( 7 k ) π 7 for k=1,2,3 we get:
( sin π 7 sin 2 π 7 sin 3 π 7 ) 2 = k = 1 6 sin k π 7 = 7 2 6
where the last equality is the case n=7 in this question: Prove that k = 1 n 1 sin k π n = n 2 n 1
So we get the desired result: sin π 7 sin 2 π 7 sin 3 π 7 = 7 8

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