Calculus help

394

Part 1

Consider the complex numbers $v$ and $w$ in the complex plane below:
$[asy] size(250);import TrigMacros;real big = 6;for (int i = 1; i < big+1; ++i) { draw(Circle((0,0),i), gray+ linewidth(0.4));} for(int i=0;i<360;i+=15) { draw(rotate(i)*((-big,0)--(big,0)),gray+ linewidth(0.4));} rr_cartesian_axes(-big,big,-big,big,complexplane=true);pair V, WW;V= 2*dir(75);WW = 4*dir(135);dot($ Say that the complex number $v, w$ and $v+w$ satisfy
$v= r_1 e^{i\theta_1}, w = r_2 e^{i\theta_2}, v+w = r_3 e^{i\theta_3},$ where $r_1, r_2, r_3$ are positive. Enter
$r_1, r_2, r_3$ in that order.

Part 2

Consider the complex numbers $v$ and $w$ in the complex plane below:
$[asy] size(250);import TrigMacros;real big = 5;for (int i = 1; i < big+1; ++i) { draw(Circle((0,0),i), gray+ linewidth(0.4));} for(int i=0;i<360;i+=15) { draw(rotate(i)*((-big,0)--(big,0)),gray+ linewidth(0.4));} rr_cartesian_axes(-big,big,-big,big,complexplane=true);pair V, WW;V= 2*dir(75);WW = 4*dir(135);dot($ Say that
$v-w = r_1 e^{i\theta_1},$ where $r_1$ is positive and $\theta_1$ is an angle in radians between $0$ and $2\pi$ . Then enter
$r_1, \theta_1$ in that order.