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Question : 9 of 160
Marks:
+1,
-0
Solution:
sin‌‌‌−i‌cos‌ ‌‌=−i(cos‌+i‌sin‌) ‌‌=−iei‌ ‌‌‌(sin‌−i‌cos‌)=−ei‌‌‌...‌ (i) ‌ ∵‌‌Z7−1=0 has roots
Z=ei‌,k=0,1,2,3,4,5,6 ∴‌‌ei‌=0 ⇒‌‌ [Sum of roots of unity is 0
] ⇒‌‌ei‌=0 ‌‌ t
e6=−1‌ ‌‌‌=−i(cos‌+i‌sin‌) ‌‌=−iei‌ ‌(sin‌−i‌cos‌)=−ei‌‌‌...‌ (i) ‌ ‌∵‌‌Z7−1=0 ‌ has roots ‌Z=ei‌,k=0,1,2,3,4,5,6 ∴‌‌ei‌=0‌‌[‌ Sum of roots of unity is ‌0] ⇒‌‌1+ei‌=0 ⇒‌‌t‌ei‌=−1 From Eq. (i),
(sin‌−i‌cos‌)=(−i)(−1)=i
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