6 find each of the following in rectangular form x iy Soluti

6.

find each of the following in rectangular form x+ iy

Solution

Given that sin(i) has to be expressed to rectangular form

sin(i) can be written as sin(0+i) =sin(x+iy)

with real part x=0 ; y=sin(i)

clearly from above we can say it is a point on y-axis(imaginary axis) where its x coordinate(real axis) is zero=>(0,sin(i)) =>is a vector joining (0,0) to (0, sin(i))

So point is on y-axis in between 1st and 2nd quardrants

let Z= 0+sin(i) can be written as => sin(i)==> (eⁱ⁽ⁱ⁾ - e^(i)-i)/2

re^i(theta) = r(cos(theta)+ i sin(theta)) by putting value of theta we will get rectangular form

1/2eⁱ⁽ⁱ⁾ = 1/2(cos(i)+isin(i))

-1/2 e^(i)-i = -1/2(cos(-i)+isin(-i))

sum of both gives 0+isin(i) is the rectangular form