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