IXL - Convert complex numbers from rectangular to polar form (Precalculus practice)

Complete the polar form of

Simplify all radicals and fractions. Express the argument

𝜃

in radians, with

≤

𝜃

cos

𝑖

sin

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Learn with an example

question

Complete the polar form of

Simplify all radicals and fractions. Express the argument

𝜃

in radians, with

≤

𝜃

cos

𝑖

sin

key idea

The modulus or absolute value

of a complex number

𝑖

is the distance between the origin and the point

𝑖

in the complex plane.

𝑖

The argument

𝜃

of the nonzero complex number

𝑖

is the counterclockwise angle from the positive real axis to the segment between the origin and the point

𝑖

Arguments are usually chosen to be between 0 and

When

the point

𝑖

is on the imaginary axis and

𝜃

When

≠

𝜃

satisfies:

tan

(

𝜃

)

The point

𝑖

has rectangular coordinates

(

)

It has polar coordinates

(

𝜃

)

Complex numbers are usually written in rectangular

form,

𝑖

The polar or trigonometric form of

𝑖

is:

[

cos

(

𝜃

)

𝑖

sin

(

𝜃

)

]

solution

The point

(

–

0) represents the complex number

–

0𝑖

–

in the complex plane. So, the rectangular form of the given number

–

Find

The modulus

is the distance between the origin and the point

–

in the complex plane.

The distance between the origin and the point is 6. So,

Find

𝜃

The argument

𝜃

is the counterclockwise angle from the positive real axis to the segment between the origin and the point

–

To find the argument, it is helpful to view the graph in polar coordinates.

Real axis

Imaginary axis

The terminal side of

𝜃

is along the negative real axis, so

𝜃

Find the polar form.

Use the values of

and

𝜃

to write the polar form.

[

cos

(

𝜃

)

𝑖

sin

(

𝜃

)

]

[

cos

(

)

𝑖

sin

(

)

]

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X.1Find the modulus and argument of a complex number

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