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The Trigonometric Form of
Complex Numbers
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Concept 1. The Trigonometric Form of Complex Numbers
C ONCEPT
1
The Trigonometric Form of
Complex Numbers
Learning Objectives
• Understand the relationship between the rectangular form of complex numbers and their corresponding polar
form.
• Convert complex numbers from standard form to polar form and vice versa.
√
A number in the form a + bi, where a and b are real numbers, and i is the imaginary unit, or −1, is called a
complex number. Despite their names, complex numbers and imaginary numbers have very real and significant
applications in both mathematics and in the real world. Complex numbers are useful in pure mathematics, providing
a more consistent and flexible number system that helps solve algebra and calculus problems. We will see some of
these applications in the examples throughout this lesson.
The Trigonometric or Polar Form of a Complex Number
The following diagram will introduce you to the relationship between complex numbers and polar coordinates.
In the figure above, the point that represents the number x + yi was plotted and a vector was drawn from the origin to
this point. As a result, an angle in standard position, θ, has been formed. In addition to this, the point that represents
x + yi is r units from the origin. Therefore, any point in the complex plane can be found if the angle θ and the r−
value are known. The following equations relate x, y, r and θ.
x = r cos θ
y = r sin θ
r2 = x2 + y2
tan θ =
y
x
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If we apply the first two equations to the point x + yi the result would be:
x + yi = r cos θ + ri sin θ → r(cos θ + i sin θ)
The right side of this equation r(cos θ + i sin θ) is called the polar or trigonometric form of a complex number. A
shortened version of this polar form is written as r cis θ. The length r is called the absolute value or the modulus,
and the angle θ is called the argument of the complex number. Therefore, the following equations define the polar
form of a complex number:
r 2 = x 2 + y2
tan θ =
y
x
x + yi = r(cos θ + i sin θ)
It is now time to implement these equations perform the operation of converting complex numbers in standard form
to complex numbers in polar form. You will use the above equations to do this.
Example 1: Represent the complex number 5 + 7i graphically and express it in its polar form.
Solution: As discussed in the Prerequisite Chapter, here is the graph of 5 + 7i.
Converting to polar from rectangular, x = 5 and y = 7.
r=
p
52 + 72 = 8.6
tan θ =
7
5
tan−1 (tan θ) = tan−1
θ = 54.5◦
So, the polar form is 8.6(cos 54.5◦ + i sin 54.5◦ ).
2
7
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Concept 1. The Trigonometric Form of Complex Numbers
Another widely used notation for the polar form of a complex number is r6 θ = r(cos θ + i sin θ). Now there are three
ways to write the polar form of a complex number.
x + yi = r(cos θ + i sin θ)
x + yi = rcisθ
x + yi = r6 θ
Example 2: Express the following polar form of each complex number using the shorthand representations.
a) 4.92(cos 214.6◦ + i sin 214.6◦ )
b) 15.6(cos 37◦ + i sin 37◦ )
Solution:
a) 4.926 214.6◦
4.92 cis 214.6◦
b) 15.66 37◦
15.6 cis 37◦
Example 3: Represent the complex number −3.12 − 4.64i graphically and give two notations of its polar form.
Solution: From the rectangular form of −3.12 − 4.64i x = −3.12 and y = −4.64
p
x2 + y2
q
r = (−3.12)2 + (−4.64)2
r=
r = 5.59
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y
x
−4.64
tan θ =
−3.12
θ = 56.1◦
tan θ =
This is the reference angle so now we must determine the measure of the angle in the third quadrant. 56.1◦ + 180◦ =
236.1◦
One polar notation of the point −3.12 − 4.64i is 5.59(cos 236.1◦ + i sin 236.1◦ ). Another polar notation of the point
is 5.596 236.1◦
So far we have expressed all values of theta in degrees. Polar form of a complex number can also have theta expressed
in radian measure. This would be beneficial when plotting the polar form of complex numbers in the polar plane.
The answer to the above example −3.12 − 4.64i with theta expressed in radian measure would be:
tan θ =
−4.64
−3.12
tan θ = .9788(reference angle)
0.9788 + 3.14 = 4.12 rad.
5.59(cos 4.12 + i sin 4.12)
Now that we have explored the polar form of complex numbers and the steps for performing these conversions, we
will look at an example in circuit analysis that requires a complex number given in polar form to be expressed in
standard form.
Example 4: The impedance Z, in ohms, in an alternating circuit is given by Z = 46506 − 35.2◦ . Express the value
for Z in standard form. (In electricity, negative angles are often used.)
Solution: The value for Z is given in polar form. From this notation, we know that r = 4650 and θ = −35.2◦ Using
these values, we can write:
Z = 4650(cos(−35.2◦ ) + i sin(−35.2◦ ))
x = 4650 cos(−35.2◦ ) → 3800
y = 4650 sin(−35.2◦ ) → −2680
Therefore the standard form is Z = 3800 − 2680i ohms.
Points to Consider
• Is it possible to perform basic operations on complex numbers in polar form?
• If operations can be performed, do the processes change for polar form or remain the same as for standard
form?
Review Questions
1. Express the following polar forms of complex numbers in the two other possible ways.
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Concept 1. The Trigonometric Form of Complex Numbers
a. 5 cis π6
b. 36 135◦
2π
c. 2 cos 2π
3 + i sin 3
2. Express the complex number 6 − 8i graphically and write it in its polar form.
3. Express the following complex numbers in their polar form.
a.
b.
c.
d.
4 + 3i
−2 + 9i
7−i
−5 − 2i
4. Graph the complex number 3(cos π4 + i sin π4 ) and express it in standard form.
5. Find the standard form of each of the complex numbers below.
a. 2 cis π2
b. 46 5π
6
c. 8 cos − π3 + i sin − π3
Review Answers
a. 5 cis π6 = 56 π6 = 5 cos π6 + i sin π6
b. 36 135◦ = 3cis135◦ = 3(cos 135◦ + i sin 135◦ )
2π
2π
6 2π
c. 2 cos 2π
3 + i sin 3 = 2cis 3 = 2 3
2. 6 − 8i
1.
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6 − 8i
x = 6 and y = −8
p
r = x2 + y2
q
r = (6)2 + (−8)2
y
x
−8
tan θ =
6
tan θ =
θ = −53.1◦
r = 10
Since θ is in the fourth quadrant then θ = −53.1◦ +360◦ = 306.9◦ Expressed in polar form 6−8i is 10(cos 306.9◦ +
i sin 306.9◦ ) or 106 306.9◦
a. 4 + 3i → x = 4, y = 3
r=
p
3
42 + 32 = 5, tan θ = → θ = 36.87◦ → 5(cos 36.87◦ + i sin 36.87◦ )
4
b. −2 + 9i → x = −2, y = 9
q
√
9
r = (−2)2 + 92 = 85 ≈ 9.22, tan θ = − → θ = 102.53◦ → 9.22(cos 102.53◦ + i sin 102.53◦ )
2
c. 7 − i → x = 7, y = −1
r=
p
√
1
72 + 12 = 50 ≈ 7.07, tan θ = − → θ = 351.87◦ → 7.07(cos 351.87◦ + i sin 351.87◦ )
7
d. −5 − 2i → x = −5, y = −2
q
√
2
r = (−5)2 + (−2)2 = 29 ≈ 5.39, tan θ = → θ = 201.8◦ → 5.39(cos 201.8◦ + i sin 201.8◦ )
5
3.
4.
5.
6.
7.
8.
9.
10.
11.
Note: The range of a graphing calculator’s
tan−1
function is limited to Quadrants I and IV, and for points located in the other quadrants, such as
−2 + 9i
in part b (in Quadrant II), you must add
180◦
to get the correct angle
θ
for numbers given in polar form.
12.
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Concept 1. The Trigonometric Form of Complex Numbers
π
π
3 cos + i sin
4
4
r=3
√
π
2
x = cos =
4 √2
2
π
y = sin =
4
2
π
4
The standard form of the polar complex number 3 cos + i sin
π
4
√
√
2
3 2
is 2 + 2 i.
3
= 2i
a. 2cis π2 → x = cos π2 = 0, y = sin π2 = 1 → 2(0) + 2(1i)
√
√
3
3 + 4 i 1 = −2 √3 + 2i
5π
5π
1
b. 46 5π
6 → x = cos 6 = − 2 , y = sin 6 = 2 → 4 − 2
2
√ √
1
3
π
π
π
π
1
c. 8 cos − 3 + i sin − 3 → x = cos − 3 = 2 , y = sin − 3 = − 2 → 8 2 + 8 − 2 3 i = 4 −
√
4i 3
7