Question

A complex number z_1z 1 ​ z, start subscript, 1, end subscript has a magnitude |z_1|=13∣z 1 ​ ∣=13vertical bar, z, start subscript, 1, end subscript, vertical bar, equals, 13 and an angle \theta_1=315^{\circ}θ 1 ​ =315 ∘ theta, start subscript, 1, end subscript, equals, 315, degrees. Express z_1z 1 ​ z, start subscript, 1, end subscript in rectangular form, as z_1=a+biz 1 ​ =a+biz, start subscript, 1, end subscript, equals, a, plus, b, i. Express a+bia+bia, plus, b, i in exact terms. z_1 =z 1 ​ =z, start subscript, 1, end subscript, equals

Answer 1

Answer:

$z_1 = \sqrt{84.5} - i \, sin(\sqrt{84.5})$

Step-by-step explanation:

$z_1$ has the following properties:

$|z_1| = 13$

$\theta = 315$ º = 1.75 π (π = 180º)

therefore, z₁ = 13 * [ cos(1.75 π) + i sin(1.75 π) ] =  13*[cos(1.75π - 2π) + i sin(1.75π - 2π)] = 13*[cos(-π/4) + i sin(-π/4)] {this is due to periodicity} = 13*(√2/2 - i  √2/2) = $\sqrt{84.5} - i \, sin(\sqrt{84.5})$