The value of 1 - sin2 -9 -icos2 -91 - sin2-9-icos2 -9 3 is-

( 1 + sin2 π/9 +icos2 π/91 + sin2π/9 icos2 π/9 )^3 = ( 1 +cos(π/2 2π/9 )+ isin(π/2 2π/9 )1 + cos(π/2 2π/9 ) i sin(π/2 2π/9 ) )^3 = ( 1 + cos(π/2 2π/ ...

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Standard XII

Mathematics

Allied Angles

The value of ...

Question

The value of ${⎛ ⎝ \frac{1 + sin \frac{2 π}{9} + i cos \frac{2 π}{9}}{1 + sin \frac{2 π}{9} - i cos \frac{2 π}{9}} ⎞ ⎠}^{3}$ is:

- \frac{1}{2} (1 - i \sqrt{3})

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\frac{1}{2} (1 - i \sqrt{3})

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- \frac{1}{2} (\sqrt{3} - i)

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\frac{1}{2} (\sqrt{3} - i)

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Solution

The correct option is C $- \frac{1}{2} (\sqrt{3} - i)$
${⎛ ⎝ \frac{1 + sin \frac{2 π}{9} + i cos \frac{2 π}{9}}{1 + sin \frac{2 π}{9} - i cos \frac{2 π}{9}} ⎞ ⎠}^{3}$
$= {⎛ ⎜ ⎜ ⎝ \frac{1 + cos (\frac{π}{2} - \frac{2 π}{9}) + i sin (\frac{π}{2} - \frac{2 π}{9})}{1 + cos (\frac{π}{2} - \frac{2 π}{9}) - i sin (\frac{π}{2} - \frac{2 π}{9})} ⎞ ⎟ ⎟ ⎠}^{3}$
$= {⎛ ⎜ ⎜ ⎝ \frac{1 + cos (\frac{π}{2} - \frac{2 π}{9}) + i sin (\frac{π}{2} - \frac{2 π}{9})}{1 + cos (\frac{π}{2} - \frac{2 π}{9}) - i sin (\frac{π}{2} - \frac{2 π}{9})} ⎞ ⎟ ⎟ ⎠}^{3}$
$= ⎛ ⎝ \frac{1 + cos \frac{5 π}{18} + i sin \frac{5 π}{18}}{1 + cos \frac{5 π}{18} - i sin \frac{5 π}{18}} ⎞ ⎠$
$= ⎛ ⎜ ⎜ ⎝ \frac{2 cos \frac{5 π}{36} {cos \frac{5 π}{36} + i sin \frac{5 π}{36}}}{2 cos \frac{5 π}{36} {cos \frac{5 π}{36} + i sin \frac{5 π}{36}}} ⎞ ⎟ ⎟ ⎠$
$= {⎛ ⎜ ⎝ \frac{e^{i (\frac{5 π}{36})}}{e^{- i (\frac{5 π}{36})}} ⎞ ⎟ ⎠}^{3}$
$= {(e^{i (\frac{5 π}{36}) + i (\frac{5 π}{36})})}^{3}$
$= {(e^{2 i (\frac{5 π}{36})})}^{3}$
$= e^{i (\frac{30 π}{36})}$
$= e^{i (\frac{5 π}{6})}$
$= - \frac{\sqrt{3}}{2} + \frac{i}{2}$

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Similar questions

{sin}^{2} \frac{π}{18} + {sin}^{2} \frac{π}{9} + {sin}^{2} \frac{7 π}{18} + {sin}^{2} \frac{4 π}{9} = k

. Find the value of

k

Q. The value

\sum_{r = 1}^{s} (s i n \frac{2 r π}{9} + i c o s \frac{2 r π}{9})

of is

The value of $s i n^{2} (\frac{π}{18}) + s i n^{2} (\frac{π}{9}) + s i n^{2} (\frac{7 π}{18}) + s i n^{2} (\frac{4 π}{9})$ is

Prove that: $s i n^{2} \frac{π}{18} + s i n^{2} \frac{π}{9} + s i n^{2} \frac{7 π}{18} + s i n^{2} \frac{4 π}{9} = 2$

Q. sin²π/18 + sin²π/9 + sin² 7π/18 + sin² 4π/9 =
(a) 1
(b) 4
(c) 2
(d) 0

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The value of ⎛⎝1+sin2π9+icos2π91+sin2π9−icos2π9⎞⎠3 is:

The value of ${⎛ ⎝ \frac{1 + sin \frac{2 π}{9} + i cos \frac{2 π}{9}}{1 + sin \frac{2 π}{9} - i cos \frac{2 π}{9}} ⎞ ⎠}^{3}$ is: