Question

1. Express the following complex numbers in cartesian form x + jy: 1/2 e jπ , 1/2 e −jπ , √ 2e j9π/4 , √ 2e −j9π/4

2. Express the following complex numbers in polar form rejθ with −π < θ ≤ π: 5, −2, 1/2 − j √ 3/2, (1 + j)/(1 − j).

PLEASE ANSWER WITH FULL STEPS AND CORRECT.

Answer 1

Answer :- 1) We use Euler's expressioin i.e. ae^jx = a*(cos(x) + jsin(x))

0.5e^j*pie = 0.5*(cos(pie) + jsin(pie)) = -0.5 + j0

0.5e^-j*pie = 0.5*(cos(pie) - jsin(pie)) = -0.5 - j0

(2)^0.5 * e^j*9pie/4 = (2)^0.5 * (cos(9*pie/4) + jsin(9*pie/4)) = (2)^0.5 *((2)^-0.5 + j(2)^-0.5 ) = 1 + j

(2)^0.5 * e^-j*9pie/4 = (2)^0.5 * (cos(9*pie/4) - jsin(9*pie/4)) = (2)^0.5 *((2)^-0.5 - j(2)^-0.5 ) = 1 - j

Answer :- 2) For a complex number x + jy, magnitude = (x² + y²)^0.5 and phase angle = tan^-1(b/a). The sign of phase angle changes as per the quadrent in which complex number lies.

5 = 5 + j0, thus magitude = 5, phase angle = 0. Hence in polar form it is 5e^j0

-2 = -2 + j0, thus magitude = 2, phase angle = 0 or pie. Since -2 lies lies on -ve real axis, so phase angle is 180 deg. or pie. Hence in polar form it is 2e^j*pie .

0.5 - j((3)^0.5/2) so magitude = 1, phase angle = tan^-1((3)^0.5) = 60⁰ = pie/3. But this number lies in 4th quadrent. Thus phase angle will be -pie/3. Hence in polar form it is 1e^-j*pie/3 .

(1 + j)/(1 - j) = (1+j)*(1+j)/((1-j)*(1+j)) = 2j/2 = j = 0 + j1, so the magitude = 1, phase angle = tan^-1(1/0) = 90⁰ = pie/2. Hence in polar form it is 1e^j*pie/2 .

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