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Trigonometric Ratios of Multiples of an Angle

cosπ11cos2π11...

Question

$cos \frac{π}{11} cos \frac{2 π}{11} cos \frac{3 π}{11} cos \frac{4 π}{11} cos \frac{5 π}{11} = \frac{1}{32}$

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Solution

$cos \frac{3 π}{11} = - cos \frac{8 π}{11}$

$cos \frac{5 π}{11} = - cos \frac{16 π}{11}$

$cos \frac{π}{11} cos \frac{2 π}{11} cos \frac{3 π}{11} cos \frac{4 π}{11} cos \frac{5 π}{11}$

$= cos \frac{π}{11} cos \frac{2 π}{11} cos \frac{8 π}{11} cos \frac{4 π}{11} cos \frac{16 π}{11}$

$= \frac{2 sin \frac{π}{11}}{2 sin \frac{π}{11}} cos \frac{π}{11} cos \frac{2 π}{11} cos \frac{8 π}{11} cos \frac{4 π}{11} cos \frac{16 π}{11}$

$= \frac{sin \frac{2 π}{11}}{2 sin \frac{π}{11}} cos \frac{2 π}{11} cos \frac{8 π}{11} cos \frac{4 π}{11} cos \frac{16 π}{11}$

$= \frac{sin \frac{4 π}{11}}{4 sin \frac{π}{11}} cos \frac{4 π}{11} cos \frac{8 π}{11} cos \frac{16 π}{11}$

$= \frac{sin \frac{8 π}{11}}{8 sin \frac{π}{11}} cos \frac{8 π}{11} cos \frac{16 π}{11}$

$= \frac{sin \frac{16 π}{11}}{16 sin \frac{π}{11}} cos \frac{16 π}{11}$

$= \frac{sin \frac{32 π}{11}}{32 sin \frac{π}{11}}$

$= \frac{sin \frac{10 π}{11}}{32 sin \frac{π}{11}}$

$= \frac{1}{32}$

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Similar questions

c o s \frac{π}{11} c o s \frac{2 π}{11} c o s \frac{3 π}{11} c o s \frac{4 π}{11} c o s \frac{5 π}{11}

Q. Find the value of -

cos \frac{π}{11} cos \frac{2 π}{11} cos \frac{3 π}{11} cos \frac{4 π}{11} cos \frac{5 π}{11}

.

Q. Prove that :
(i)

c o s \frac{2 π}{7} c o s \frac{4 π}{7} c o s \frac{6 π}{7} = \frac{1}{8}

c o s \frac{π}{11} c o s \frac{2 π}{11} c o s \frac{3 π}{11} c o s \frac{4 π}{11} c o s \frac{5 π}{11} = \frac{1}{32}

c o s \frac{π}{11} + c o s \frac{3 π}{11} + c o s \frac{5 π}{11} + c o s \frac{7 π}{11} + c o s \frac{9 π}{11} = \frac{1}{2}

Q. Using eular's formula prove that

\frac{c o s π}{11} + \frac{c o s 3 π}{11} + \frac{c o 5 π}{11} + \frac{c o s 7 π}{11} + \frac{c o s 9 π}{11} = \frac{1}{2} s i n x

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