Evaluate 1 iiSolutionz 1ii z rcosx i sinx Let rcosx 1 a

Evaluate (1 + i)^i

Solution

z = (1+i)^i

z = r(cosx + i sinx),

Let,
r.cosx = 1,
and r.sinx = 1,
Then squaring and adding, we get,
r^2.(cosx^2 + sinx^2) = 1 + 1 = 2
r^2 = 2
r = sqrt(2) ........................... [1]
Also, r.sinx / r.cosx = 1/1 = 1
tanx = 1
x = pi/4 = 45 degree ................... [2]

(1 + i)^i = (r^i).(cosx + i.sinx)^i

= (sqrt(2))^i.(cosix + i.sinix)

  = (sqrt(2))^i.[cos(i.pi/4) + i.sin(i.pi/4)]