what-is-the-value-of-cos-i-j-with-i-2-1-and-j-e-i-2pi-3-

Question Number 35223 by abdo mathsup 649 cc last updated on 16/May/18

w h a t i s t h e v a l u e o f c o s (i + j) w i t h i^{2} = - 1 a n d

j = e^{i \frac{2 π}{3}} ?

Answered by sma3l2996 last updated on 17/May/18

c o s (i + j) = \frac{e^{(i + j) i} + e^{- (i + j) i}}{2} = \frac{e^{- 1 + i j} + e^{1 - i j}}{2}

i j = i \times e^{i \frac{2 π}{3}} = i (\frac{1}{2} - i \frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{2} + i \frac{1}{2}

c o s (i + j) = \frac{e^{- 1 + \frac{\sqrt{3}}{2} + i \frac{1}{2}} + e^{1 - \frac{\sqrt{3}}{2} - i \frac{1}{2}}}{2} = \frac{e^{\frac{\sqrt{3}}{2} - 1} \times e^{i / 2} + \frac{e^{- i / 2}}{e^{\sqrt{3} / 2 - 1}}}{2}

= \frac{1}{2 e^{\sqrt{3} / 2 - 1}} (e^{\sqrt{3} - 2} (c o s \frac{1}{2} + i s i n \frac{1}{2}) + c o s \frac{1}{2} - i s i n \frac{1}{2})

\frac{1}{2 e^{\sqrt{3} / 2 - 1}} ((e^{\sqrt{3} - 2} + 1) c o s (1 / 2) + i (e^{\sqrt{3} - 2} - 1) s i n (1 / 2))

Commented by abdo imad last updated on 17/May/18

s i r S m a 3 l y o u r m e t h o d i s c o r r e c t b u t y o u h a v e c o m m i t e d

a s m a l l e r r o r e^{i \frac{2 π}{3}} = - \frac{1}{2} + i \frac{\sqrt{3}}{2}! \dots