Convert the polar representation of this complex number into its rectangular form: z=5(cos pi+i sin pi)

The complex number z = -5 into its rectangular form

Step-by-step explanation:

* Lets revise the complex numbers

- If z = r(cos Ф ± i sin Ф), where r cos Ф is the real part and i r sin Ф is the

 imaginary part in the polar form

- The value of i = √(-1) ⇒ imaginary number

- Then z = a + bi , where a is the real part and bi is the imaginary part

  in the rectangular form

∴ a = r cos Ф and b = r sin Ф

* Lets solve the problem

∵ z = r (cos Ф ± i sin Ф)

∵ z = 5 (cos π + i sin π)

∴ The real part is 5 cos π

∴ The imaginary part is 5 sin π

- Lets find the values of cos π and sin π

∵ The angle of measure π is on the negative part of x axis at the

  point (-1 , 0)

∵ x = cos π and y = sin π

∴ cos π = -1  

∴ sin π = 0

∴ a = 5(-1) = -5

∴ b = 5(0) = 0

∴ z = -5 + i (0)

* The complex number z = -5 into its rectangular form

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step-by-step explanation: