IXL - Find the modulus and argument of a complex number (Precalculus practice)

Find the modulus

and argument

𝜃

Simplify all radicals and fractions. Express the argument

𝜃

in radians, with

≤

𝜃

ref_doc_title.

Learn with an example

question

Find the modulus

and argument

𝜃

Simplify all radicals and fractions. Express the argument

𝜃

in radians, with

≤

𝜃

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, 6) represents the complex number

6𝑖

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

6𝑖

Find

The modulus

is the distance between the origin and the point

6𝑖

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

6𝑖

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 positive imaginary axis, so

𝜃

Looking for more? Try this:

$Confetti$

Excellent!

Now entering the Challenge Zone—are you ready?

Questions
answeredQuestions

Time
elapsedTime

hr	min	sec
00	00	04

SmartScore
out of 100
IXL's SmartScore is a dynamic measure of progress towards mastery, rather than a percentage grade. It tracks your skill level as you tackle progressively more difficult questions. Consistently answer questions correctly to reach excellence (90), or conquer the Challenge Zone to achieve mastery (100)! Learn more

Teacher tools

Work it out

Not feeling ready yet? These can help:

W.6Absolute value in the complex plane

HH.3Graphs of angles

R.10Inverses of trigonometric functions using a calculator

Excellent! You got that right!

Jumping to level 1 of 1

Excellent!