R = C_1/C_2," where " C_1 = 4 -8i" and " C_2 = 9 + iR=C1C2, where C1=4−8i and C2=9+i
Convert each complex number to its polar form:
C_1 = |C_1|/_theta_1 and " C_2 = |C_2| |/_theta_2C1=|C1|∠θ1andC2=|C2|∣∠θ2
Then, R = |C_1|/|C_2| /_theta_1-theta_2 R=|C1||C2|∠θ1−θ2
C_1=sqrt(4^2 + (-8)^2)=sqrt(80); C_2=sqrt(9^2 + (1)^2)=sqrt(82) C1=√42+(−8)2=√80;C2=√92+(1)2=√82
theta_1 = tan^-1 (-2); theta_2 = tan^-1 (1/9); θ1=tan−1(−2);θ2=tan−1(19);
theta_R=theta_1-theta_2 =tan^-1 (-2) - tan^-1 (1/9) = -1.218 " rads"θR=θ1−θ2=tan−1(−2)−tan−1(19)=−1.218 rads
Thus R = .975 /_(theta_R = -1.218)R=.975∠(θR=−1.218) this is now in the polar form
R=r/_theta_R; r = .975 " the magnitude of " R, theta_R=-69.775^oR=r∠θR;r=.975 the magnitude of R,θR=−69.775o
Now you can convert back to the rectangular coordinate
R = r_x + r_y; R_x = |r|costheta_R + |r|sintheta_RR=rx+ry;Rx=|r|cosθR+|r|sinθR
r_x = .975cos69.8 ~~.337; r_y=.975sin(-69.8) ~~-.915 rx=.975cos69.8≈.337;ry=.975sin(−69.8)≈−.915
R ~~.337 + -915i R≈.337+−915i
