Search in book...
Toggle Font Controls
Create new playlist

Name your new playlist

Playlist description (optional)
Sign In

Email address

Password

Forgot Password?

or

Continue with Facebook

Continue with Google
Sign Up

Full Name

Email address

Confirm Email Address

Password

or

Continue with Facebook

Continue with Google

12.4 Polar Form of a Complex Number

Polar Form $r (\cos θ + j \sin θ)$ $r (\cos θ + j \sin θ)$ • Expressing Numbers in Polar Form

In this section, we use the fact that a complex number can be represented by a vector to write complex numbers in another form. This form has advantages when multiplying and dividing complex numbers, and we will discuss these operations later in this chapter.

In the complex plane, by drawing a vector from the origin to the point that represents the number $x + y j,$ $x + y j,$ an angle $θ$ $θ$ in standard position is formed. The point $x + y j$ $x + y j$ is r units from the origin. In fact, we can find any point in the complex plane by knowing the angle $θ$ $θ$ and the value of r. The equations relating x, y, r, and $θ$ $θ$ are similar to those developed for vectors in Chapter 9. Referring to Fig. 12.9, we see that

Position vector of length r goes to x plus y j at counterclockwise angle theta. Horizontal length is x = r times cosine of theta, and vertical length is y = r times sine of theta. — Fig. 12.9

\begin{array}{rcl} x & = & r \cos θ & y & = & r \sin θ \end{array}

$\begin{array}{rcl} x & = & r \cos θ & y & = & r \sin θ \end{array}$ (12.7)

\begin{array}{rcl} r^{2} & = & x^{2} + y^{2} & tan θ = \frac{y}{x} \end{array}

$\begin{array}{rcl} r^{2} & = & x^{2} + y^{2} & tan θ = \frac{y}{x} \end{array}$ (12.8)

Substituting Eq. (12.7) into the rectangular form $x + y j$ $x + y j$ of a complex number, we have $x + y j = r \cos θ + j (r \sin θ),$ $x + y j = r \cos θ + j (r \sin θ),$ or

x + y j = r (\cos θ + j \sin θ)

$x + y j = r (\cos θ + j \sin θ)$ (12.9)

The right side of Eq. (12.9) is called the polar form (sometimes the trigonometric form). The length r is the absolute value, or the modulus, and $θ$ $θ$ is the argument of the complex number. Equations (12.7)–(12.9) define the polar form of a complex number.

EXAMPLE 1 Representing a number in polar form

Represent the complex number $3 + 4 j$ $3 + 4 j$ graphically and give its polar form.

From the rectangular form $3 + 4 j,$ $3 + 4 j,$ we see that $x = 3$ $x = 3$ and $y = 4.$ $y = 4.$ Using Eqs. (12.8), we have

r = \sqrt{3^{2} + 4^{2}} = 5 θ = \tan^{- 1} \frac{4}{3} = 53.1 °

$r = \sqrt{3^{2} + 4^{2}} = 5 θ = \tan^{- 1} \frac{4}{3} = 53.1 °$

Thus, the polar form is $5 (\cos 53.1 ° + j \sin 53.1 °) .$ $5 (\cos 53.1 ° + j \sin 53.1 °) .$ See Fig. 12.10.

A position vector of length r = 5 goes to 3 plus 4 j at (3, 4) at counterclockwise angle 53.1 degrees. — Fig. 12.10

In Example 1, in expressing the polar form as $5 (\cos 53.1 ° + j \sin 53.1 °),$ $5 (\cos 53.1 ° + j \sin 53.1 °),$ we rounded the angle to the nearest $0.1 °$ $0.1 °$ , as it is not possible to express the result exactly in degrees. In dealing with nonexact numbers, we will express angle to the nearest $0.1 °$ $0.1 °$ . Other results that cannot be written exactly will be expressed to three significant digits, unless a different accuracy is given in the problem. Of course, in applied situations, most numbers are approximate, as they are derived through measurement.

Another convenient and widely used notation for the polar form is $r ∠ \underline{θ} .$ $r ∠ \underline{θ} .$ We must remember in using this form that it represents a complex number and is simply a shorthand way of writing $r (\cos θ + j \sin θ) .$ $r (\cos θ + j \sin θ) .$ Therefore,

r ∠ \underline{θ} = r (\cos θ + j \sin θ)

$r ∠ \underline{θ} = r (\cos θ + j \sin θ)$ (12.10)

EXAMPLE 2 Polar form r ∠ θ_

$3 (\cos 40 ° + j \sin 40 °) = 3 ∠ \underline{40 °}$ $3 (\cos 40 ° + j \sin 40 °) = 3 ∠ \underline{40 °}$
$6.26 (\cos 217.3 ° + j \sin 217.3 °) = 6.26 ∠ \underline{217.3 °}$ $6.26 (\cos 217.3 ° + j \sin 217.3 °) = 6.26 ∠ \underline{217.3 °}$
$5 ∠ \underline{120 °} = 5 (\cos 120 ° + j \sin 120 °)$ $5 ∠ \underline{120 °} = 5 (\cos 120 ° + j \sin 120 °)$
$14.5 ∠ \underline{306.2 °} = 14.5 (\cos 306.2 ° + j \sin 306.2 °)$ $14.5 ∠ \underline{306.2 °} = 14.5 (\cos 306.2 ° + j \sin 306.2 °)$

EXAMPLE 3 Rectangular form to polar form

Represent the complex number $- 1.04 - 1.56 j$ $- 1.04 - 1.56 j$ graphically and give its polar form.

The graphical representation is shown in Fig. 12.11. From Eqs. (12.8), we have

A position vector goes to (negative 1.04, negative 1.56) at counterclockwise angle 236.3 degrees. — Fig. 12.11

\begin{array}{rcl} r & = & \sqrt{{(- 1.04)}^{2} + {(- 1.56)}^{2}} = 1.87 \\ θ_{ref} & = & \tan^{- 1} \frac{1.56}{1.04} = 56.3 ° θ = 180 ° + 56.3 ° = 236.3 ° \end{array}

$\begin{array}{rcl} r & = & \sqrt{{(- 1.04)}^{2} + {(- 1.56)}^{2}} = 1.87 \\ θ_{ref} & = & \tan^{- 1} \frac{1.56}{1.04} = 56.3 ° θ = 180 ° + 56.3 ° = 236.3 ° \end{array}$

Because both the real and imaginary parts are negative, $θ$ $θ$ is a third-quadrant angle. Therefore, we found the reference angle before finding $θ .$ $θ .$ This means the polar forms are

1.87 (\cos 236.3 ° + j \sin 236.3 °) = 1.87 ∠ \underline{236.3 °}

$1.87 (\cos 236.3 ° + j \sin 236.3 °) = 1.87 ∠ \underline{236.3 °}$

EXAMPLE 4 Polar form to rectangular form—impedance

The impedance Z (in $Ω$ $Ω$ ) in an alternating-current circuit is given by $Z = 3560 ∠ \underline{- 32.4 °} .$ $Z = 3560 ∠ \underline{- 32.4 °} .$ Express this in rectangular form.

From the polar form, we have $r = 3560 Ω$ $r = 3560 Ω$ and $θ = - 32.4 °$ $θ = - 32.4 °$ (it is common to use negative angles in this type of application). This means that we can also write

Z = 3560 [(\cos (- 32.4 °) + j \sin (- 32.4 °)]

$Z = 3560 [(\cos (- 32.4 °) + j \sin (- 32.4 °)]$

See Fig. 12.12. This means that

A position vector goes to Z in quadrant 4 at clockwise angle negative 32.4 degrees. — Fig. 12.12

\begin{array}{rcl} x & = & 3560 \cos (- 32.4 °) = 3010 \\ y & = & 3560 \sin (- 32.4 °) = - 1910 \end{array}

$\begin{array}{rcl} x & = & 3560 \cos (- 32.4 °) = 3010 \\ y & = & 3560 \sin (- 32.4 °) = - 1910 \end{array}$

Therefore, the rectangular form is $Z = 3010 - 1910 j Ω .$ $Z = 3010 - 1910 j Ω .$

Complex numbers can be converted between rectangular and polar forms on most calculators. Figure 12.13 shows the calculator conversions for Examples 3 and 4. Note that for Example 3, the calculator returns an angle of $- 123.7 ° .$ $- 123.7 ° .$ This is coterminal to $236.3 °$ $236.3 °$ , the angle we found.

Figure 12.13 Full Alternative Text

EXAMPLE 5 Polar form—real and imaginary numbers

Represent the numbers 5, $- 5,$ $- 5,$ 7j, and $- 7 j$ $- 7 j$ in polar form. See Fig. 12.14.

Four numbers are represented on a plane. Five is at (5, 0), 7 j is (0, 7), negative 5 is (negative 5, 0), and negative 7 j is (0, negative 7). — Fig. 12.14

Real numbers lie on the real axis in the complex plane, while imaginary numbers lie on the imaginary axis. Therefore, the polar form angle of any real or imaginary number is a multiple of $90 °$ $90 °$ as shown below:

\begin{array}{rcll} 5 & = & 5 (\cos 0 ° + j \sin 0 °) = 5 ∠ \underline{0 °} & 5 units from the origin at an angle of 0 ° \\ - 5 & = & 5 (\cos 180 ° + j \sin 180 °) = 5 ∠ \underline{180 °} & 5 units from the origin at an angle of 180 ° \\ 7 j & = & 7 (\cos 90 ° + j \sin 90 °) = 7 ∠ \underline{90 °} & 7 units from the origin at an angle of 90 ° \\ - 7 j & = & 7 (\cos 270 ° + j \sin 270 °) = 7 ∠ \underline{270 °} & 7 units from the origin at an angle of 270 ° \end{array}

$\begin{array}{rcll} 5 & = & 5 (\cos 0 ° + j \sin 0 °) = 5 ∠ \underline{0 °} & 5 units from the origin at an angle of 0 ° \\ - 5 & = & 5 (\cos 180 ° + j \sin 180 °) = 5 ∠ \underline{180 °} & 5 units from the origin at an angle of 180 ° \\ 7 j & = & 7 (\cos 90 ° + j \sin 90 °) = 7 ∠ \underline{90 °} & 7 units from the origin at an angle of 90 ° \\ - 7 j & = & 7 (\cos 270 ° + j \sin 270 °) = 7 ∠ \underline{270 °} & 7 units from the origin at an angle of 270 ° \end{array}$

EXERCISES 12.4

In Exercises 1 and 2, change the sign of the real part of the complex number in the indicated example of this section and then perform the indicated operations for the resulting complex number.

Example 1
Example 3

In Exercises 3–18, represent each complex number graphically and give the polar form of each.

$8 + 6 j$ $8 + 6 j$
$- 8 - 15 j$ $- 8 - 15 j$
$30 - 40 j$ $30 - 40 j$
$12 j - 5$ $12 j - 5$
$3.00 j - 2.00$ $3.00 j - 2.00$
$7.00 - 5.00 j$ $7.00 - 5.00 j$
$- 0.55 - 0.24 j$ $- 0.55 - 0.24 j$
$460 - 460 j$ $460 - 460 j$
$1 + j \sqrt{3}$ $1 + j \sqrt{3}$
$\sqrt{2} - j \sqrt{2}$ $\sqrt{2} - j \sqrt{2}$
$3.514 - 7.256 j$ $3.514 - 7.256 j$
$95.27 j + 62.31$ $95.27 j + 62.31$
$- 3$ $- 3$
60
9j
$- 2 j$ $- 2 j$

In Exercises 19–36, represent each complex number graphically and give the rectangular form of each.

$5.00 (\cos 54.0 ° + j \sin 54.0 °)$ $5.00 (\cos 54.0 ° + j \sin 54.0 °)$
$6 (\cos 180 ° + j \sin 180 °)$ $6 (\cos 180 ° + j \sin 180 °)$
$160 (\cos 150.0 ° + j \sin 150.0 °)$ $160 (\cos 150.0 ° + j \sin 150.0 °)$
$2.50 (\cos 315.0 ° + j \sin 315.0 °)$ $2.50 (\cos 315.0 ° + j \sin 315.0 °)$
$3.00 (\cos 232.0 ° + j \sin 232.0 °)$ $3.00 (\cos 232.0 ° + j \sin 232.0 °)$
$220.8 (\cos 155.13 ° + j \sin 155.13 °)$ $220.8 (\cos 155.13 ° + j \sin 155.13 °)$
$0.08 (\cos 360 ° + j \sin 360 °)$ $0.08 (\cos 360 ° + j \sin 360 °)$
$15 (\cos 0 ° + j \sin 0 °)$ $15 (\cos 0 ° + j \sin 0 °)$
$120 (\cos 270 ° + j \sin 270 °)$ $120 (\cos 270 ° + j \sin 270 °)$
$\cos 600.0 ° + j \sin 600.0 °$ $\cos 600.0 ° + j \sin 600.0 °$
$4.75 ∠ \underline{172.8 °}$ $4.75 ∠ \underline{172.8 °}$
$1.50 ∠ \underline{897.7 °}$ $1.50 ∠ \underline{897.7 °}$
$0.9326 ∠ \underline{229.54 °}$ $0.9326 ∠ \underline{229.54 °}$
$277.8 ∠ \underline{- 342.63 °}$ $277.8 ∠ \underline{- 342.63 °}$
$7.32 ∠ \underline{- 270 °}$ $7.32 ∠ \underline{- 270 °}$
$18.3 ∠ \underline{540.0 °}$ $18.3 ∠ \underline{540.0 °}$
$86.42 ∠ \underline{94.62 °}$ $86.42 ∠ \underline{94.62 °}$
$4629 ∠ \underline{182.44 °}$ $4629 ∠ \underline{182.44 °}$

In Exercises 37–44, solve the given problems.

What is the argument for any negative real number?
For $x + y j,$ $x + y j,$ what is the argument if $x = y < 0 ?$ $x = y < 0 ?$
Show that the conjugate of $r ∠ \underline{θ}$ $r ∠ \underline{θ}$ is $r ∠ \underline{- θ} .$ $r ∠ \underline{- θ} .$
The impedance in a certain circuit is $Z = 8.5 ∠ \underline{- 36 °} Ω .$ $Z = 8.5 ∠ \underline{- 36 °} Ω .$ Write this in rectangular form.
The voltage of a certain generator is represented by $2.84 - 1.06 j kV .$ $2.84 - 1.06 j kV .$ Write this voltage in polar form.
Find the magnitude and direction of a force on a bolt that is represented by $40.5 + 24.5 j$ $40.5 + 24.5 j$ newtons.
The electric field intensity of a light wave can be described by $12.4 ∠ \underline{78.3 °} V / m .$ $12.4 ∠ \underline{78.3 °} V / m .$ Write this in rectangular form.
The current in a certain microprocessor circuit is represented by $3.75 ∠ \underline{15.0 °} μ A .$ $3.75 ∠ \underline{15.0 °} μ A .$ Write this in rectangular form.

Answers to Practice Exercises

$17 (\cos 331.9 ° + j \sin 331.9 °)$ $17 (\cos 331.9 ° + j \sin 331.9 °)$
$- 1.25 + 2.17 j$ $- 1.25 + 2.17 j$
$10 ∠ \underline{270 °}$ $10 ∠ \underline{270 °}$

..................Content has been hidden....................

You can't read the all page of ebook, please click here login for view all page.

Table of Contents for
12.4 Polar Form of a Complex Number

12.4 Polar Form of a Complex Number

Fig. 12.9

EXAMPLE 1 Representing a number in polar form

Fig. 12.10

EXAMPLE 2 Polar form r ∠ θ_

EXAMPLE 3 Rectangular form to polar form

Fig. 12.11

EXAMPLE 4 Polar form to rectangular form—impedance

Fig. 12.12

Fig. 12.13

EXAMPLE 5 Polar form—real and imaginary numbers

Fig. 12.14

EXERCISES 12.4

Answers to Practice Exercises

Table of Contents for 12.4 Polar Form of a Complex Number

Create new playlist

Sign In

Sign Up

Table of Contents for
12.4 Polar Form of a Complex Number