Maths > Complex Numbers > 5.0 Representation of complex number
Complex Numbers
1.0 Definition
2.0 Algebraic operations
3.0 Conjugate of complex number
4.0 Properties of conjugate
5.0 Representation of complex number
5.1 Cartesian form (Geometric Representation)
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
6.0 Euler's formulae
7.0 Properties of Argument
8.0 De Moivre's Theorem
9.0 Square root of a complex number
10.0 The ${n^{th}}$ root of unity
11.0 Cube roots of unity
12.0 Rotation
13.0 Geometrical properties
14.0 Locus
15.0 Ptolemy's Theorem
5.4 Trigonometric/Polar Representation
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
Let $z=a+ib$, $z$ is represented by a point $P(a,b)$ in the argand plane. By geometrical representation,
$$\left| z \right| = OP = \sqrt {{a^2} + {b^2}} $$ and $$\arg (z) = \angle PON$$
In $\Delta OPM$,
$$\begin{equation} \begin{aligned} a = OP\cos \left( {\angle PON} \right) = \left| z \right|\cos (\arg z)\quad ...(1) \\ b = OP\sin \left( {\angle PON} \right) = \left| z \right|\sin (\arg z)\quad ...(2) \\ \because z = a + ib,{\text{ assumed above}} \\\end{aligned} \end{equation} $$
From $(1)$ and $(2)$, $$z = r\left( {\cos \theta + isin\theta } \right)$$
In the above expression, always take principal value of $\theta $.