Question

Find the value of $\sum\limits_{k=1}^{8}{\left[ \left( \dfrac{\cos 2k\pi }{9}+i\dfrac{\sin 2k\pi }{9} \right) \right]}$

Answer 1

Open up the summation for each value of k(i.e. from 1 to 8) and compare it with the expression for the 9th root of unity, as given below, to obtain the answer.
$1+w+{{w}^{2}}+{{w}^{3}}+{{w}^{4}}+{{w}^{5}}+{{w}^{6}}+{{w}^{7}}+{{w}^{8}}=0$

Complete step-by-step answer:
$\sum\limits_{k=1}^{8}{\left[ \left( \dfrac{\cos 2k\pi }{9}+i\dfrac{\sin 2k\pi }{9} \right) \right]}$
We know that 9th roots of unit are $1+w+{{w}^{2}}+{{w}^{3}}+{{w}^{4}}+{{w}^{5}}+{{w}^{6}}+{{w}^{7}}+{{w}^{8}}$
Sum of roots
$1+w+{{w}^{2}}+{{w}^{3}}+{{w}^{4}}+{{w}^{5}}+{{w}^{6}}+{{w}^{7}}+{{w}^{8}}=0$
$w+{{w}^{2}}+{{w}^{3}}+{{w}^{4}}+{{w}^{5}}+{{w}^{6}}+{{w}^{7}}+{{w}^{8}}=-1$
$\therefore \text{ if k=1}$

& \cos \dfrac{2\left( 1 \right)\pi }{9}+i\text{ }\sin \dfrac{2\left( 1 \right)\pi }{9} \\\ & =\cos \dfrac{2\pi }{9}+\text{ }i\text{ }\sin \dfrac{2\pi }{9} \\\ & =w \\\ \end{aligned}$$ If $\begin{aligned} & k=2 \\\ & \\\ \end{aligned}$ $$\begin{aligned} & \cos \dfrac{2\left( 2 \right)\pi }{9}+i\text{ }\sin \dfrac{2\left( 2 \right)\pi }{9} \\\ & =\cos \dfrac{4\pi }{9}+\text{ }i\text{ }\sin \dfrac{4\pi }{9} \\\ & ={{w}^{2}} \\\ \end{aligned}$$ If $k=3$ then ${{w}^{3}}$ If $k=4$ then ${{w}^{4}}$ If $k=5$ then ${{w}^{5}}$ If $k=6$ then ${{w}^{6}}$ If $k=7$ then ${{w}^{7}}$ If $k=8$ then ${{w}^{8}}$ Solved as, $$\begin{aligned} & \cos \dfrac{2\left( 8 \right)\pi }{9}+i\text{ }\sin \dfrac{2\left( 8 \right)\pi }{9} \\\ & =\cos \dfrac{16\pi }{9}+\text{ }i\text{ }\sin \dfrac{16\pi }{9} \\\ & ={{w}^{8}} \\\ \end{aligned}$$ The sum of all the terms $$\sum\limits_{k=1}^{8}{\left[ \cos \left( \dfrac{2k\pi }{9} \right)+i\text{ }\sin \left( \dfrac{2k\pi }{9} \right) \right]}=-1$$ **Note:** $1,w\text{ and }{{\text{w}}^{2}}$ represents the cube roots of units and also $1+w+{{w}^{2}}=0$ Some important properties of cube roots of units are: Property 1: Among the three cube roots of unity one of the cube roots is real and the other two are conjugate complex numbers. Property 2: Square of any one imaginary cube root of unity is equal to the other imaginary cube root of unity.