Question

A complex number z_1z 1 ​ z, start subscript, 1, end subscript has a magnitude |z_1|=6∣z 1 ​ ∣=6vertical bar, z, start subscript, 1, end subscript, vertical bar, equals, 6 and an angle \theta_1=70^{\circ}θ 1 ​ =70 ∘ theta, start subscript, 1, end subscript, equals, 70, 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. Round aaa and bbb to the nearest thousandth.

Answer 1

Answer:

Step-by-step explanation:

Given the modulus of a complex number |z1| = 6 and its argument θ1 = 70°

If z1 = a+bi, according to definition;

|z1| = √a²+b²

|z1|² = a²+b²

Since |z1| = 6, |z1|² = 6² = 36

a²+b² = 36 .......... 1

Also the argument is derived from the expression

tan θ = b/a

tan 70 = b/a

b/a = 2.75

b = 2.75a ........ 2

Substitute equation 2 into 1

a²+b² = 36

a²+(2.75a)² = 36

a²+7.5625a² = 36

8.5625a² = 36

a² = 36/8.5625

a² = 4.204379

a = √4.204379

a = 2.051 (to nearest thousandth)

Since b = b = 2.75a

b = 2.75(2.051)

b = 5.639 (to nearest thousandth)

z1 = 2.051+5.639i