Basic roots of unity properties

3 In this question, \(w\) denotes the complex number \(\cos \frac { 2 } { 5 } \pi + \mathrm { i } \sin \frac { 2 } { 5 } \pi\).

1. Express \(w ^ { 2 } , w ^ { 3 }\) and \(w ^ { * }\) in polar form, with arguments in the interval \(0 \leqslant \theta < 2 \pi\).
2. The points in an Argand diagram which represent the numbers $$1 , \quad 1 + w , \quad 1 + w + w ^ { 2 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } , \quad 1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 }$$ are denoted by \(A , B , C , D , E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.)
3. Write down a polynomial equation of degree 5 which is satisfied by \(w\).

9

1. Write down the five fifth roots of unity.
2. Hence find all the roots of the equation $$z ^ { 5 } + 16 + ( 16 \sqrt { } 3 ) i = 0$$ giving answers in the form \(r \mathrm { e } ^ { \mathrm { i } q \pi }\), where \(r > 0\) and \(q\) is a rational number. Show these roots on an Argand diagram. Let \(w\) be a root of the equation in part (ii).
3. Show that $$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
4. Identify the root for which \(| 2 - w |\) is least.

3

1. Write down the fifth roots of unity.
2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).

6 Write down all the 8th roots of unity. Verify that $$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$ Hence express \(z ^ { 8 } - 1\) as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.

State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{94ecfc6e-df52-45a0-8f7b-f33fda391b15-4_903_883_477_502}

1. Justify each of the following statements.
   (a) \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   (b) \(A B C D E\) is a regular pentagon.
   (c) \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   (d) \(b ^ { * } = e\)
   (e) \(a + b + c + d + e = 0\)
2. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

9 In this question, the argument of a complex number is defined as being in the range \([ 0,2 \pi )\).
You are given that \(\omega _ { k }\), where \(k = 0,1,2 , \ldots , n - 1\), are the \(n n ^ { \text {th } }\) roots of unity for some integer \(n , n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega _ { 0 } = 1\) ).

1. With the help of a diagram explain why \(\omega _ { k } = \left( \omega _ { 1 } \right) ^ { k }\) for \(k = 2 , \ldots , n - 1\).
2. Using the identity given in part (a), show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \omega _ { \mathrm { k } } = 0\).
3. Show that if \(z\) is a complex number then \(z + z ^ { * } = 2 \operatorname { Re } ( z )\).
4. Using the results from parts (b) and (c) show that \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } \operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = 0\).
5. With the help of a diagram explain why \(\operatorname { Re } \left( \omega _ { \mathrm { k } } \right) = \operatorname { Re } \left( \omega _ { \mathrm { n } - \mathrm { k } } \right)\) for \(k = 1,2 , \ldots , n - 1\). You should now consider the case where \(n = 5\).
6. 1. Use parts (d) and (e) to deduce that \(\cos \frac { 4 \pi } { 5 } = \mathrm { a } + \mathrm { b } \cos \frac { 2 \pi } { 5 }\), for some rational constants \(a\) and \(b\).
   2. Hence determine the exact value of \(\cos \frac { 2 \pi } { 5 }\).

10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\).
\includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}

1. Justify each of the following statements.
   (a) \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
   (b) \(A B C D E\) is a regular pentagon.
   (c) \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
   (d) \(b ^ { * } = e\)
   (e) \(a + b + c + d + e = 0\)
2. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).

12 The Argand diagram shows the solutions to the equation \(z ^ { 5 } = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-22_1079_995_354_520} 12

1. Solve the equation $$z ^ { 5 } = 1$$ giving your answers in the form \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(0 \leq \theta < 2 \pi\)
   [0pt] [2 marks] 12
2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}]{a889963c-266c-497e-b7fc-99a249ba9e58-23_2484_1726_219_141}
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3. The Argand diagram on page 22 is repeated below.
   \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-24_1079_1000_354_520} Explain, with reference to the Argand diagram, why the expression $$16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1$$ has a repeated quadratic factor.
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4. \(O\) is the centre of a regular pentagon \(A B C D E\) such that \(O A = O B = O C = O D =\) \(O E = 1\) unit.
   The distance from \(O\) to \(A B\) is \(h\)
   By solving the equation \(16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1 = 0\), show that $$h = \frac { \sqrt { 5 } + 1 } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-26_2492_1721_217_150}