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. z_1 =z 1 ​ =z, start subscript, 1, end subscript, equals + ii

Answer 1

The answer is "
`\bold{9.19- 9.19\ i}`"

Explanation:

When the value of
`z_1` has the following properties:

`\to |z_1| =13`

`\theta = 315^(\circ) \\\\ = 1.75 \pi\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ _(where) \ \ (\pi = 180^(\circ))`

Calculating the value of
`z_1` :

`= 13 * [ \cos(1.75 \pi ) + i \sin(1.75 \pi) ] \\\\= 13 *[\cos(1.75 \pi - 2\pi ) + \ i \sin(1.75 \pi - 2\pi )]\\ \\= 13 * [\cos(-0.25 \pi ) +\ i \sin(-0.25 \pi) ]\\\\= 13 * [0.707106781 - 0.707106781]\\\\ =9.19 - 9.19\ i \\\\`