Determine and examine the 𝑛th roots of unity and their location on the unit circle.

The equation \(z^6 = 1\) has six solutions in the complex plane. (a) Determine all six roots in polar form \(r\operatorname{cis}(\theta)\) where \(0 \leq \theta < 2\pi\). (b) Express the root with the smallest positive argument in Cartesian form \(a + bi\), giving exact values.

Consider the equation \(z^6 = 1\) where \(z \in \mathbb{C}\). (a) Determine all six roots in polar form \(r\operatorname{cis}(\theta)\) where \(0 \leq \theta < 2\pi\). (2 marks) (b) Express the root with the smallest positive argument in Cartesian form \(a + bi\). (1 mark) (c) Determine the product of all roots that lie in the upper half-plane (that is, where \(\operatorname{Im}(z) > 0\)). Express your answer in Cartesian form. (2 marks)

Determine the five fifth roots of unity and express each root in the form \(r\operatorname{cis}(\theta)\) where \(0 \leq \theta < 2\pi\). Hence, determine the sum of all five roots.

Consider the equation \(z^6 = 1\) where \(z \in \mathbb{C}\). (a) Determine all six roots of this equation in Cartesian form \(a + bi\) where \(a, b \in \mathbb{R}\). (3 marks) (b) Hence determine the sum of all sixth roots of unity that lie in the upper half of the complex plane (that is, where \(\text{Im}(z) > 0\)). Give your answer in exact form. (2 marks)

Which of the following complex numbers is a fifth root of unity?

Consider the equation \(z^6 = 1\). (a) Determine all solutions to this equation in the form \(z = \text{cis}(\theta)\) where \(0 \leq \theta < 2\pi\). (2 marks) (b) Express the solution with the smallest positive argument in Cartesian form \(a + bi\), where \(a\) and \(b\) are exact values. (2 marks)

Determine the five fifth roots of unity in the form \(z = \cos(\theta) + i\sin(\theta)\), where \(0 \le \theta < 2\pi\).

Consider the equation \(z^6 = 1\). (a) Determine all six solutions in modulus-argument form \(r\operatorname{cis}(\theta)\), where \(0 \leq \theta < 2\pi\). (3 marks) (b) Hence determine the sum of all sixth roots of unity that lie in the upper half of the complex plane (that is, where \(\operatorname{Im}(z) > 0\)). Give your answer in Cartesian form \(a + bi\). (2 marks)

Find all cube roots of unity and express each in Cartesian form \(a + bi\), where \(a\) and \(b\) are exact values.

Determine the five fifth roots of unity in the form \(z = \cos\theta + i\sin\theta\) where \(0 \leq \theta < 2\pi\).