What is the value of the expression sin &#x2061;<!-- ⁡ --> 2 &#x03C0;<!-- π

excluderho

excluderho

Answered question

2022-06-23

What is the value of the expression sin 2 π 7 sin 4 π 7 + sin 4 π 7 sin 8 π 7 + sin 8 π 7 sin 2 π 7

Answer & Explanation

Stevinivm

Stevinivm

Beginner2022-06-24Added 18 answers

As you observed,
sin α sin β = 1 2 [ cos ( α β ) cos ( α + β ) ]
Applying that observation here yields
sin ( 2 π 7 ) sin ( 4 π 7 ) + sin ( 4 π 7 ) sin ( 8 π 7 ) + sin ( 8 π 7 ) sin ( 2 π 7 ) = 1 2 [ cos ( 2 π 7 ) cos ( 6 π 7 ) + cos ( 4 π 7 ) cos ( 12 π 7 ) + cos ( 6 π 7 ) cos ( 10 π 7 ) ] = 1 2 [ cos ( 2 π 7 ) + cos ( 4 π 7 ) cos ( 12 π 7 ) cos ( 10 π 7 ) ] = 1 2 [ cos ( 2 π 7 ) + cos ( 4 π 7 ) cos ( 12 π 7 ) cos ( 10 π 7 ) ] = 1 2 [ cos ( 2 π 7 ) + cos ( π 3 π 7 ) cos ( 2 π 2 π 7 ) cos ( π + 3 π 7 ) ] = 1 2 [ cos ( 2 π 7 ) cos ( 3 π 7 ) cos ( 2 π 7 ) + cos ( 3 π 7 ) ] = 0
where we have used the identities
cos ( θ ) = cos θ cos ( π θ ) = cos θ cos ( π + θ ) = cos θ cos ( 2 π θ ) = cos θ

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