The complex number in polar form (with θ expressed in radians).?

Question

a) The complex number z = −1 + i in polar form (with θ expressed in radians) is z = _________. The complex number z = 2cos π/6 + i sin π/6 in rectangular form is z =_________. 

b) The complex number graphed below can be expressed in rectangular form as z =_________ or in polar form (with θ expressed in radians) as 
z =_________. 

Answer 1

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ), where r = | z | = √(a² + b²), a = r cos θ, and b = r sin θ, and θ = tan^{- 1}(b / a) for a > 0 or θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o for a < 0.

• a).

The complex number is z = - 1 + i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here a = - 1 < 0 and b = 1.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (- 1)² + (1)² ]

            = √[ 1 + 1 ]

            = √2.

Now find the argument θ.

Since a = - 1 < 0, use the formula θ = tan^{- 1}(b / a) + π.

θ = tan^{- 1}[ 1/(- 1) ] + π

θ = tan^{- 1}(- 1) + π

θ = tan^{- 1}(tan(π/2 + π/4)) + π

θ = 3π/4 + π

θ = 7π/4.

Note that here θ is measured in radians.

Therefore, the polar form of - 1 + i is about  √2[ cos (7π/4) + i sin (7π/4) ].

 

The complex number z = 2cos (π/6) + i sin (π/6).

Rectangular form of the complex number z is z = a + ib.

Substitute the values cos (π/6) = √3/2 and sin (π/6) = 1/2.

z = 2(√3/2) + i (1/2)

⇒ z = √3 + (0.5)i.

Therefore, the rectangular form of z is √3 + (0.5)i.

Answer 2

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ),

where, r = | z | = √(a² + b²),

a = r cos θ,

b = r sin θ, and

if a > 0, then θ = tan^{- 1}(b / a), and

if a < 0, then θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o.

• a).

The complex number is z = - 1 + i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here a = - 1 < 0 and b = 1.

So, first find the absolute value of r .

r = | z | = √(a² + b²)

            = √[ (- 1)² + (1)² ]

            = √[ 1 + 1 ]

            = √2.

Now find the argument θ.

Since a = - 1 < 0, use the formula θ = tan^{- 1}(b / a) + π.

θ = tan^{- 1}[ 1/(- 1) ] + π

θ = tan^{- 1}(- 1) + π

θ = tan^{- 1}(tan(π/2 + π/4)) + π

θ = 3π/4 + π

θ = 7π/4.

Note that here θ is measured in radians.

Therefore, the polar form of - 1 + i is about  √2[ cos (7π/4) + i sin (7π/4) ].

 

The complex number z = 2cos (π/6) + i sin (π/6).

Rectangular form of the complex number z is z = a + ib.

Substitute the values cos (π/6) = √3/2 and sin (π/6) = 1/2.

z = 2(√3/2) + i (1/2)

⇒ z = √3 + (0.5)i.

Therefore, the rectangular form of z is √3 + (0.5)i.