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

Write

cos

𝑖

sin

in rectangular form.

Simplify any radicals.

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Learn with an example

question

Write

cos

𝑖

sin

in rectangular form.

Simplify any radicals.

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.

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

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

)

]

a
=
r
cos
(
𝜃
)
is the real part
b
=
r
sin
(
𝜃
)
is the imaginary part

solution

The complex number

cos

𝑖

sin

has modulus 1 and argument

so it has polar coordinates

Look at its graph.

Real axis

Imaginary axis

The point with polar coordinates

represents 𝑖 in the complex plane, so

cos

𝑖

sin

𝑖

You can check this by evaluating cosine and sine.

cos

𝑖

sin

𝑖

$Confetti$

Excellent!

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Not feeling ready yet? These can help:

X.2Convert complex numbers from rectangular to polar form

II.10Find trigonometric functions using a calculator

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