If A=cos2π33       sin2π33−sin2π33     cos2π33 then   A2017=

# If

1. A

$A$

2. B

${A}^{2}$

3. C

${A}^{3}$

4. D

${A}^{4}$

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### Solution:

$Let\text{\hspace{0.17em}\hspace{0.17em}}\theta =\frac{2\pi }{33}⇒A=\left[\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$

$⇒{A}^{2017}=\left[\begin{array}{cc}\mathrm{cos}2017\text{\hspace{0.17em}}\theta & \mathrm{sin}2017\text{\hspace{0.17em}}\theta \\ -\mathrm{sin}2017\text{\hspace{0.17em}}\theta & \mathrm{cos}2017\text{\hspace{0.17em}}\theta \end{array}\right]$

$2017\text{\hspace{0.17em}}\theta =\left(2017\right)\frac{2\pi }{33}=122\pi +\frac{8\pi }{33}⇒\mathrm{cos}\left(2017\text{\hspace{0.17em}}\theta \right)=\mathrm{cos}4\text{\hspace{0.17em}}\theta ;$

$\mathrm{sin}\left(2017\theta \right)=\mathrm{sin}4\theta$  +91

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