A2 October 2021 Paper 1 Q10

(i) Find the exact roots of the equation \(z^3 - 1 = \sqrt{3}\,\mathrm{i}\), expressing them in the form \(r\mathrm{e}^{\mathrm{i}\theta}\), where \(r \gt 0\) and \(-\pi \lt \theta \lt \pi\). [5]

(ii) The points representing the cube roots of unity form a triangle \(\Delta_1\). The points representing the roots of the equation \(z^3 - 1 = \sqrt{3}\,\mathrm{i}\) form a triangle \(\Delta_2\).
State a sequence of two transformations that maps \(\Delta_1\) onto \(\Delta_2\). [2]

(iii) The three roots in part (b)(i) are \(z_1\), \(z_2\) and \(z_3\).
By simplifying \(z_1 + z_2 + z_3\), verify that the sum of these roots is zero. [2]

(iv) Hence show that \(\sin 20^\circ + \sin 140^\circ = \sin 100^\circ\). [2]