Roots of Unity

Roots of Unity: Level 4 Challenges


On the complex plane, the vertices of a regular polygon centred at (0,0)(0,0) are the geometric images of the nthn^\text{th} roots of unity. The area of the polygon is 3. How many sides does it have?

cos(π11)cos(2π11)+cos(3π11)cos(4π11)+cos(5π11)= ?\cos\left( \frac{\pi}{11}\right) - \cos\left( \frac{2\pi}{11} \right)+ \cos\left( \frac{3\pi}{11}\right) \\ - \cos\left( \frac{4\pi}{11} \right)+ \cos\left( \frac{5\pi}{11} \right)=\ ?

Given z2+z+1=0,z^2+z+1=0, find the value of

(z+1z)2+(z2+1z2)2+(z3+1z3)2++(z21+1z21)2.\left(z+\dfrac{1}{z} \right)^2+\left(z^2+\dfrac{1}{z^2}\right)^2+\left(z^3+\dfrac{1}{z^3} \right)^2+\cdots+\left(z^{21}+\dfrac{1}{z^{21}}\right)^2.

312δ112δ1+312δ212δ2+312δ312δ3 \dfrac{31-2\delta_{1}}{1-2\delta_{1}} +\dfrac{31-2\delta_{2}}{1-2\delta_{2}}+\dfrac{31-2\delta_{3}}{1-2\delta_{3}}

If 1,δ1,δ2,δ31,\delta_{1},\delta_{2},\delta_{3} are distinct fourth roots of unity, then evaluate the expression above.

If w=e2πi/5w=e^{2\pi{i}/5}, find k=1411+wk+w2k\sum_{k=1}^{4}\frac{1}{1+w^k+w^{2k}}

If for zC ,z \in \mathbb{C} \space , z+1z=2cos6°.z+\dfrac{1}{z}=2 \cos 6°. Then find the value of (z1000+1z1000).\left( z^{1000}+\frac{1}{z^{1000}} \right).


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