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

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) = [tex] \sqrt{84.5} - i \, sin(\sqrt{84.5}) [/tex]

keshavgandhi04 keshavgandhi04 · Answer 2 · 2021-11-28T12:13:25+01:00

The complex number is [tex]\rm z_1 = 9.2-9.2i[/tex] and this can be determined by using the rectangular form of [tex]z_1[/tex] and also using the given data.

Given :

[tex]|z_1| = 13[/tex]

[tex]\theta_1 = 315^\circ[/tex]

The rectangular form of [tex]z_1[/tex] is given by:

[tex]\rm z_1 = a+bi[/tex]

[tex]\rm |z_1|=\sqrt{a^2+b^2}[/tex]

[tex]\rm 13^2=a^2+b^2[/tex] ---- (1)

Now, the argument of [tex]z_1[/tex] is given by:

[tex]\rm \theta_1=tan^{-1}\dfrac{a}{b}[/tex]

[tex]\rm tan(315^\circ)=tan(tan^{-1}\dfrac{a}{b})[/tex]

[tex]\rm \dfrac{b}{a}=tan(315^\circ)[/tex]

b = -a

Now, substitute the value of 'b' in equation (1).

[tex]2a^2 = 169[/tex]

a = 9.2

b = -9.2

So, the complex number is [tex]\rm z_1 = 9.2-9.2i[/tex] and this can be determined by using the rectangular form of [tex]z_1[/tex] and also using the given data.

For more information, refer to the link given below:

https://brainly.com/question/23017717