Polar form of complex numbers is another way of representing complex number.
z= a+ib is the rectangular form of a complex number.
z = r(cos θ + i sin θ) is the polar form of the complex number.
Here 'r' represents the absolute value and 'θ' represents argument of the complex number.
In the above diagram x- axis is the real axis and y-axis is the imaginary axis. We will express real and imaginary part of the complex number in the form of r and θ where r is the modulus and θ is the argument of the complex number. θ is the angle made by the complex number with the real ( x) axis.
In the above diagram, we know that
r² = a² + b²
By the trigonometric ratios we know that
cos θ = a/r and sin θ = b/r.
we can rewrite it as
r cosθ = a and r sinθ = b.
Substituting the value of a and b in the rectangular coordinate form we get
z = r cosθ + i r sinθ
z = r(cosθ + i sinθ)
where r = ∣z∣ = √(a² + b²) and
θ= tan ⁻¹ (b/a)
Note: If a > 0 then θ= tan ⁻¹ (b/a).
If a < 0 then θ= tan ⁻¹ (b/a) + π
θ= tan ⁻¹ (b/a) + 180°
Express z= 1+i in the form of polar.
Let us find the value of r.
In the given complex number a = 1 and b = 1
So the value of r is √(a² + b²)
r = √(1² + 1²)
and cos θ = a/r and sin θ = b/r.
cos θ = 1/√2 and sin θ = 1/√2
which implies θ = tan ⁻¹ (b/a)
= tan ⁻¹ (1/1).
= tan ⁻¹ (1).
So the polar form is z = 1/√2(cos π/4 + i sin π/4).
Convert in to rectangular form.
z = 2(cos 30° + i sin 30° )
z = 2 cos30° + i 2sin 30°
a = r cosθ and b = r sin θ.
So a = 2 cos 30° and b = 2 sin 30°
a = 2(0.87) and b = 2(0.5)
a = 1.74 and b = 1
The rectangular form is z = 1.74 + i
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