if-a-1-i-2-find-a-1943-

Question Number 123607 by mohammad17 last updated on 26/Nov/20

i f a = \frac{1 + i}{\sqrt{2}} f i n d a^{1943} ?

Answered by Dwaipayan Shikari last updated on 26/Nov/20

a = \frac{1 + i}{\sqrt{2}} = \sqrt{i}

a^{1943} = {(i)}^{\frac{1943}{2}} = i^{970} . i . \sqrt{i} = - i \sqrt{i}

A n o t h e r w a y

a = \frac{1 + i}{\sqrt{2}} = e^{\frac{i π}{4}} \Rightarrow a^{1943} = e^{\frac{1943 π i}{4}} = e^{435 π i} e^{\frac{3 π}{4} i} = - 1 . (- \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}})

= \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}

Answered by malwan last updated on 26/Nov/20

a = [1, \frac{π}{4}]

\Rightarrow a^{1943} = [1^{1943}, 1943 \times \frac{π}{4}] = [1, π] [1, \frac{3 π}{4}]

= - 1 (c o s \frac{3 π}{4} + i s i n \frac{3 π}{4}) = \frac{1 - i}{\sqrt{2}}

Answered by TANMAY PANACEA last updated on 26/Nov/20

a = \frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} = c o s \frac{π}{4} + i s i n \frac{π}{4} = e^{i \times \frac{π}{4}}

1943 = 4 \times 485 + 3

a = e^{i \times \frac{π}{4}}

a^{1943} = {(e^{i \times \frac{π}{4}})}^{4 \times 485 + 3} = e^{i \times 485 π + i \times \frac{3 π}{4}}

e^{i \times 485 π} \times e^{i \times \frac{3 π}{4}}

= (c o s 485 π + i s i n 485 π) (c o s 135^{o} + i s i n 135^{o})

= {c o s (242 \times 2 π + π) + i \times 0} (- \frac{1}{\sqrt{2}} + i \times \frac{1}{\sqrt{2}})

= (c o s π + 0) (\frac{- 1}{\sqrt{2}} + i \times \frac{1}{\sqrt{2}})

= (- 1) (\frac{- 1}{\sqrt{2}} + i \times \frac{1}{\sqrt{2}}) = \frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}