Objective: by the end of this lesson you should be able to describe where a complex number is in modulus-argument form

You have described a point in the x,y plane before.
There is a different way to describe them though.
This is modulus–argument form.
This is how far away from a point and in what direction.
Modulus = how far away
Argument = angle

Work out the modulus + argument.
Put it in the following formula: z
= r (cos
θ + j sin
θ
)
Modulus is straightforward.
Argument is a bit more work, we need to use radians and some angles are greater than 90.
R must be positive, cos + sin must also be positive, ө must be the same.

For the complex number z=x+yj the argument is Tan -1 (y/x)
When you do this on your calculator it will give you the answer between -90 and 90.
This is good if the angle is in the 1 st or
4 th quadrant.

nd
rd
If the angle is in the 2 nd quadrant then add π .
If the angle is in the 3 rd quadrant then minus π .
You must also use exact answers in terms of π if they come up.

Write the following complex numbers in modulus-argument form.
6+8j

-5+12j

-3-
√
27

cos(
π
-
α
) = -cos
α cos( α π ) = -cos α cos(α ) = cos α sin(
π
-
α
) = sin
α sin( α π ) = -sin α sin (α ) = -sin α
Eh?
Use graphs of cosine and sine to help show these relationships!!

Write down the values of the modulus and argument of the following
4


 cos
π
7
+ j sin
π
7



3


 cos
π
4
− j sin
π
4


 sin (α ) = -sin α cos(-
α
) = cos
α

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