Question

A complex number z_1z 1 z, start subscript, 1, end subscript has a magnitude |z_1|=7∣z 1 ∣=7vertical bar, z, start subscript, 1, end subscript, vertical bar, equals, 7 and an angle \theta_1=270^{\circ}θ 1 =270 theta, start subscript, 1, en subscript, equals, 270, 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.

Answer 1

z = 5.785 + 6.894i

Explanation:

Answer 2

z1 = 0+7i.

Explanation:

Given the complex number

|z1| = 7 and θ1 = 270°

We are to express z1 in rectangular form as;

z1 = a + bi

|z1| = √a²+b² = 7

a²+b² = 7²

a²+b² = 49

.also argument θ1 = tan^-1(b/a) = 270°

Apply tan to both sides

tan(tan^-1(b/a)) = tan270°

b/a = tan270°

b/a = ∞

Since b/a doesn't exist, this means that the denominator is zero i.e a = 0

Substitute a = 0 into a²+b² = 49 to get b as shown;

0²+b² = 49

b² = 49

b = ±√49

b = ±7

Substitute a= 0 and b = ±7 into the rectangular form of the complex number.

z1 = a+bi

z1 = 0+7i

Hence the value of z1 in its rectangular form is z1 = 0+7i.