Represent the complex number z = 1 + i 3 in the polar form.

Represent the complex number z = 1 + i 3 in the polar form. Polar form of a complex number is z = r cos θ + i sin θ Now, z = 1 + i 3 = r cos θ + i sin θ ⇒ 1 = r cos θ , 3 = r sin θ r 2 cos 2 θ + sin 2 θ = 4 i.e., r = 4 = 2 (conventionally, r > 0 ) So, cos θ = 1 2 , sin θ = 3 2 , which gives θ = π 3 Required polar form is z = 2 cos π 3 + i sin π 3 . The complex number z = 1 + i 3 can be represented as follows. Thus, we represented z = 1 + i 3 and its polar form is z = 2 cos π 3 + i sin π 3 .

Represent the complex number z = 1 + i 3 in the polar form.

Represent the complex number z = 1 + i 3 in the polar form. Polar form of a complex number is z = r cos θ + i sin θ Now, z = 1 + i 3 = r cos θ + i sin θ ⇒ 1 = r cos θ , 3 = r sin θ r 2 cos 2 θ + sin 2 θ = 4 i.e., r = 4 = 2 (conventionally, r > 0 ) So, cos θ = 1 2 , sin θ = 3 2 , which gives θ = π 3 Required polar form is z = 2 cos π 3 + i sin π 3 . The complex number z = 1 + i 3 can be represented as follows. Thus, we represented z = 1 + i 3 and its polar form is z = 2 cos π 3 + i sin π 3 .

EASY

Earn 100

Represent the complex number $z = 1 + i \sqrt{3}$ in the polar form.

Important Questions on Complex Numbers

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

The principal value of

i^{i}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

Z_{r} = \sin \frac{2 π r}{11} - i \cos \frac{2 π r}{11},

then

\sum_{r = 0}^{10} Z_{r}

is equal to

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

Im (\frac{2 z + 1}{i z + 1}) = - 2,

then the locus of the point representing

z

in the complex plane is

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

The polar form of the complex number

\frac{1 + i}{1 - i}

is equal to

HARD

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

The real part of

e^{(3 + 4 i) x}

HARD

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

Let $i = \sqrt{- 1} .$ If $\frac{{(- 1 + i \sqrt{3})}^{21}}{{(1 - i)}^{24}} + \frac{{(1 + i \sqrt{3})}^{21}}{{(1 + i)}^{24}} = k,$ and $n = [|k|]$ be the greatest integral part of $|k| .$

Then $\sum_{j = 0}^{n + 5} {(j + 5)}^{2} - \sum_{j = 0}^{n + 5} (j + 5)$ is equal to ________.

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

The points

z_{i} = x + i y

and

z_{2} = \frac{1}{- x + i y}

in the complex plane lie on

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

a_{r} = \cos \frac{2 r π}{9} + i \sin \frac{2 r π}{9}, r = 1, 2, 3, \dots, i = \sqrt{- 1}

, then the determinant

|\begin{array}{l} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{array}|

is equal to :

HARD

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

z_{1}

and

{\bar{z}}_{1}

represents adjacent vertices of a regular polygon of

n

sides and if

\frac{I m (z_{1})}{R e (z_{1})} = \sqrt{2} - 1

, then

n

is equal to

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

{(\frac{1 + \cos (3 θ) + i \sin (3 θ)}{1 + \cos (3 θ) - i \sin (3 θ)})}^{20} = ?

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

\frac{[(\cos 3 t + i \sin 3 t)^{4} (\cos 4 t - i \sin 4 t)^{5} (\cos 5 t + isin 5 t)^{4}]}{(\cos 4 t + isin 4 t)^{3}} =

HARD

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

For any real number

r,

let

A_{r} = \{e^{i π r n} : n

is a natural number

}

be a set of complex numbers. Then

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

It is given that

z_{r} = \cos (π / 2^{r}) + i \sin (π / 2^{r})

The value of the product

z_{1} \cdot z_{2} \cdot z_{3} \cdot z_{4} \dots \infty

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

If the imaginary part of

\frac{2 z + 1}{i z + 1}

- 4,

then the locus of the point representing

z

in the complex plane is

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

\sin (i x) =

EASY

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

|z_{1}| = |z_{2}|

and

\arg (z_{1}) + \arg (z_{2}) = 0

, then

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

2 + 2 \sqrt{3} i = k (\cos θ + i \sin θ), (k > 0),

then

\frac{1}{\sqrt{3}} [\cos 6 θ + i \sin 6 θ] =

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

The general value of the real angle

θ,

which satisfies the equation,

(\cos θ + i \sin θ) (\cos 2 θ + i \sin 2 θ) \dots (\cos n θ + i \sin n θ) = 1

is given by, (assuming

k

is an integer)

MEDIUM

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

z

and

ω

are two complex numbers such that

|z ω| = 1

and

a r g (z) - a r g (ω) = \frac{π}{2}

, then:

HARD

Mathematics>Algebra>Complex Numbers>Argand Plane and Polar Representation

Let

z_{1}

and

z_{2}

be any two non-zero complex numbers such that

3 |z_{1}| = 4 |z_{2}| .

z = \frac{3 z_{1}}{2 z_{2}} + \frac{2 z_{2}}{3 z_{1}}

then maximum value of

|z|

Represent the complex number z=1+i3 in the polar form.

Important Questions on Complex Numbers

Learn and Prepare for any exam you want !

Represent the complex number $z = 1 + i \sqrt{3}$ in the polar form.