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If z_(1) and z_(2) are two of the 8^(th) roots of unity such that arg(z_(1)/z_(2)) is last positive, then z_(1)/z_(2) is |
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Answer» `1+i` ALSO, `n^(th)` roots of unity are given by `z_(R) = (e^(i2rpi)/n),r=0 1,2,…..,(n-1)` `therefore 8^(th)` roots of unity are given by `r=0,1,2,....,7` We know that arg `(z_(1)/z_(2))="arg"(z_(1))-"arg"(z_(2))`. Therefore, arg`((z_(1))/z_(2))` will have least positive value, if `z_(1)` and `z_(2)` represent any two consecutive points on the circle `|z|=1`. Let `z_(1)=z_(r)=e^((irpi)/4)`. Then, `z_(2)=z_(r-1)=e^(i(r-1)pi/4)` `therefore z_(1)/z_(2)=e^(ipi//4)=cospi/4+isinpi/4=(1+i)/sqrt(2)` |
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