Point P is uniquely determined by the ordered pair of real numbers ( r , θ ), called the polar coordinates of the point P .
Numerical : Represent the complex number z = 1 + i √3 in the polar form . Solution : let z = 1 + i √3 = r ( cos θ + i sin θ ) r =| z | = ( a 2 + b 2 ) 1 / 2 = (( 1 ) 2 + ( √3 ) 2 ) 1 / 2 = 2 Comparing real parts of z = 1 + i √3 = r ( cos θ + i sin θ ) = 2 ( cos θ + i sin θ ) 1 = 2 cos θ or cos θ = ½ or cos θ = π / 3
Therefore , polar representation will be z = r ( cos θ + i sin θ ) = 2 ( cos π / 3 + i sin π / 3 )
Quadratic Equation We have seen of real numbers in the cases where discriminant is non-negative , i . e ., ≥ 0 ,
Let us consider the following quadratic equation : ax 2 + bx + c = 0 with real coefficients a , b , c and a ≠ 0 .
Also , let us assume that the b 2 – 4ac < 0 . Numerical : Solve x 2 + x + 1 = 0 Solution : Determinant , b 2 – 4ac = 12 – 4 × 1 × 1 = 1 – 4 = – 3 X = ( -1 ± I √3 )/ 2