Trigonometric or Polar or Modulus Argument Form of a Complex Number
Let the point `P` represent the nonzero complex number `z = x + iy.` Let the directed line segment `OP` be of length `r` and `θ` be the angle which `OP` makes with the positive direction of `x`-axis `Fig (a).`
We may note that the point `P` is uniquely determined by the ordered pair of real numbers `(r, θ)`, called the `text(polar coordinates of the point P.)` We consider the origin as the pole and the positive direction of the `x` axis as the initial line.
We have, `x = r cos θ, y = r sin θ` and therefore, `z = r (cos θ + i sin θ).` The latter is said to be `text(the polar form of the complex number.)` Here `r = x^2 + y^2 = |z|` is the modulus of `z` and `θ` is called the argument (or amplitude) of `z` which is denoted by `arg z.`
For any complex number `z ≠ 0,` there corresponds only one value of `θ` in `0 ≤ θ < 2π`. However, any other interval of length `2π,` for example `- π < θ ≤ π,` can be such an interval.We shall take the value of `θ` such that `- π < θ ≤ π,` called `text(principal argument of z.)`
`text(Points to consider :)`
1. `costheta + isintheta` is also written as `CiS theta.`
2. `1 = cos 0 + i sin 0,` `1 = cospi + i sin pi`
`i=cospi/2 +isinpi/2,` `-i =cospi/2 - isinpi/2 `
We may note that the point `P` is uniquely determined by the ordered pair of real numbers `(r, θ)`, called the `text(polar coordinates of the point P.)` We consider the origin as the pole and the positive direction of the `x` axis as the initial line.
We have, `x = r cos θ, y = r sin θ` and therefore, `z = r (cos θ + i sin θ).` The latter is said to be `text(the polar form of the complex number.)` Here `r = x^2 + y^2 = |z|` is the modulus of `z` and `θ` is called the argument (or amplitude) of `z` which is denoted by `arg z.`
For any complex number `z ≠ 0,` there corresponds only one value of `θ` in `0 ≤ θ < 2π`. However, any other interval of length `2π,` for example `- π < θ ≤ π,` can be such an interval.We shall take the value of `θ` such that `- π < θ ≤ π,` called `text(principal argument of z.)`
`text(Points to consider :)`
1. `costheta + isintheta` is also written as `CiS theta.`
2. `1 = cos 0 + i sin 0,` `1 = cospi + i sin pi`
`i=cospi/2 +isinpi/2,` `-i =cospi/2 - isinpi/2 `