How do you evaluate e^( ( 7 pi)/6 i) - e^( ( 14 pi)/3 i)e7π6ie14π3i using trigonometric functions?

1 Answer
Mar 30, 2016

-(sqrt3 -1)/2 - i (sqrt3 +1)/2312i3+12

Explanation:

e^((7pi i)/6)= cos ((7pi)/6) + i sin ((7pi)/6)e7πi6=cos(7π6)+isin(7π6)

= -cos pi/6 - i sin pi/6cosπ6isinπ6 [(7pi)/6 = pi +pi/6[7π6=π+π6, In the third quadrant both sin and cos would b negative]

Like wise e^((14pi i)/3)= cos ((14pi)/3) + i sin ((14pi)/3)e14πi3=cos(14π3)+isin(14π3)

= cos ((2pi)/3) + i sin ((2pi)/3)cos(2π3)+isin(2π3) [(14pi)/3= 4pi +(2pi)/3= (2pi)/3][14π3=4π+2π3=2π3]

= - cos (pi/3) +i sin (pi/3) cos(π3)+isin(π3)

Now combining both expressions, it would be -sqrt3 /2 -i/2 +1/2 -i sqrt3 /232i2+12i32