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Find the principal argument of the compl...

Find the principal argument of the complex number `sin(6pi)/5+i(1+cos(6pi)/5)dot`

Text Solution

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`Z = sin.(6pi)/(5) + i(1 + cos .(6pi)/(5))`
`= 2 cos.(3pi)/(5)( sin .(3pi)/(5) +i cos.(3pi)/(5))`
`2 cos.(3pi)/(5)(cos((pi)/(2)-(3pi)/(5))+i sin ((pi)/(2)-(3pi)/(5))`
`=2 cos.(3pi)/(5)[cos((-pi)/(10)) + i sin ((-pi)/(10))]`
`=2 cos.(3pi)/(5)[-cos.(pi)/(10) + i sin.(pi)/(10)]`
`= - 2 cos .(3pi)/(5)[ cos(pi-(pi)/(10)) + i sin (pi-(pi)/(10))]`
`=- 2 cos.(3pi)/(5)[cos.(9pi)/(10 )+ i sin.(9pi)/(10)]`
` therefore" " arg(z) = (9pi)/(10), and |z| = - 2 cos .(3pi)/(5)`
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