Trigonometric Trigonometric Form of Complex Form of Complex Numbers 6.6aNumbers 6.6a

The first stuff in our last section of the chapter!

But first, remind me – what’s a complex number???

a biA complex number is one that can be written in the form

where a and b are real numbers. The real number a is thereal part, the real number b is the imaginary part, anda + bi is the standard form.

And of course, remember the definition of the imaginary number:

1i

In Sec. 6.1, we learned how to write a vector in“trigonometric form”:

i jv a b

cosθ i sin θ jv v v

cosθi sin θjv

Now, we will do something similarNow, we will do something similarwith with complex numberscomplex numbers……

Recall how we graph complex numbers:

ImaginaryAxis

RealAxis

P(a, b)z = a + bi

r

0a

b cosθ sin θr r i

sin θb r

tan θb

a

cosθ sin θr i

cosθa r

2 2r z a b

z a bi

Definition: Trigonometric Form of a Complex NumberThe trigonometric form of the complex number z = a + bi is

cosθ sin θz r i The number r is the absolute value or modulus of z,and 0 is an argument of z.

Is the argument of any particular complex number Is the argument of any particular complex number uniqueunique??

Practice changing forms of complex numbers

1 3i

0 θ 2π

221 3

Switch forms of the given complex number, for

1 3r i

5π 5π1 3 2cos 2 sin

3 3i i

π 5πθ 2π

3 3

π

3

2

(between trigonometric form and standard form)(between trigonometric form and standard form)

How about a graph???

Reference angle: so…

Practice changing forms of complex numbers

3 4i 0 θ 2π Switch forms of the given complex number, for

3 4 5 cos 4.069 sin 4.069i i

Practice changing forms of complex numbers

π π3 cos sin

6 6i

0 θ 2π Switch forms of the given complex number, for

π π 3 13 cos sin 3

6 6 2 2i i

In this case, simply evaluate the trigonometric functions…

3 3 3

2 2i

Practice changing forms of complex numbers

17 cos105 sin105i

0 θ 2π Switch forms of the given complex number, for

17 cos105 sin105 1.067 3.983i i

Practice changing forms of complex numbers

5i0 θ 2π Switch forms of the given complex number, for

π π5 5 cos sin

2 2i i

Whiteboard Problems:

0 θ 2π Switch forms of the given complex number, for

1. 3 3i

2. 8(cos210 sin 210 )i

3. 5(cos sin )4 4i