10.5 Polar form of complex numbers (Page 2/8)

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Given $z = 1 - 7 i,$ find $| z | .$

$| z | = \sqrt{50} = 5 \sqrt{2}$

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Writing complex numbers in polar form

The polar form of a complex number expresses a number in terms of an angle $θ$ and its distance from the origin $r .$ Given a complex number in rectangular form expressed as $z = x + y i,$ we use the same conversion formulas as we do to write the number in trigonometric form:

\begin{array}{l} x = r \cos θ \\ y = r \sin θ \\ r = \sqrt{x^{2} + y^{2}} \end{array}

We review these relationships in [link] .

Triangle plotted in the complex plane (x axis is real, y axis is imaginary). Base is along the x/real axis, height is some y/imaginary value in Q 1, and hypotenuse r extends from origin to that point (x+yi) in Q 1. The angle at the origin is theta. There is an arc going through (x+yi).

We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point $(x, y) .$ The modulus, then, is the same as $r,$ the radius in polar form. We use $θ$ to indicate the angle of direction (just as with polar coordinates). Substituting, we have

\begin{array}{l} z = x + y i \\ z = r \cos θ + (r \sin θ) i \\ z = r (\cos θ + i \sin θ) \end{array}

a general note label

Polar form of a complex number

Writing a complex number in polar form involves the following conversion formulas:

\begin{array}{l} x = r \cos θ \\ y = r \sin θ \\ r = \sqrt{x^{2} + y^{2}} \end{array}

Making a direct substitution, we have

\begin{array}{l} z = x + y i \\ z = (r \cos θ) + i (r \sin θ) \\ z = r (\cos θ + i \sin θ) \end{array}

where $r$ is the modulus and $θ$ is the argument . We often use the abbreviation $r cis θ$ to represent $r (\cos θ + i \sin θ) .$

Expressing a complex number using polar coordinates

Express the complex number $4 i$ using polar coordinates.

On the complex plane, the number $z = 4 i$ is the same as $z = 0 + 4 i .$ Writing it in polar form, we have to calculate $r$ first.

\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ r = \sqrt{0^{2} + 4^{2}} \\ r = \sqrt{16} \\ r = 4 \end{array}

Next, we look at $x .$ If $x = r \cos θ,$ and $x = 0,$ then $θ = \frac{π}{2} .$ In polar coordinates, the complex number $z = 0 + 4 i$ can be written as $z = 4 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$ or $4 cis (\frac{π}{2}) .$ See [link] .

Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi/2).

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Express $z = 3 i$ as $r cis θ$ in polar form.

$z = 3 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

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Finding the polar form of a complex number

Find the polar form of $- 4 + 4 i .$

First, find the value of $r .$

\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ r = \sqrt{{(- 4)}^{2} + (4^{2})} \\ r = \sqrt{32} \\ r = 4 \sqrt{2} \end{array}

Find the angle $θ$ using the formula:

\begin{array}{l} \cos θ = \frac{x}{r} \\ \cos θ = \frac{- 4}{4 \sqrt{2}} \\ \cos θ = - \frac{1}{\sqrt{2}} \\ θ = \cos^{- 1} (- \frac{1}{\sqrt{2}}) = \frac{3 π}{4} \end{array}

Thus, the solution is $4 \sqrt{2} cis (\frac{3 π}{4}) .$

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Write $z = \sqrt{3} + i$ in polar form.

$z = 2 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))$

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Converting a complex number from polar to rectangular form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given $z = r (\cos θ + i \sin θ),$ first evaluate the trigonometric functions $\cos θ$ and $\sin θ .$ Then, multiply through by $r .$

Converting from polar to rectangular form

Convert the polar form of the given complex number to rectangular form:

z = 12 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))

We begin by evaluating the trigonometric expressions.

\cos (\frac{π}{6}) = \frac{\sqrt{3}}{2} and \sin (\frac{π}{6}) = \frac{1}{2}

After substitution, the complex number is

z = 12 (\frac{\sqrt{3}}{2} + \frac{1}{2} i)

We apply the distributive property:

\begin{array}{l} z = 12 (\frac{\sqrt{3}}{2} + \frac{1}{2} i) \\ = (12) \frac{\sqrt{3}}{2} + (12) \frac{1}{2} i \\ = 6 \sqrt{3} + 6 i \end{array}

The rectangular form of the given point in complex form is $6 \sqrt{3} + 6 i .$

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Finding the rectangular form of a complex number

Find the rectangular form of the complex number given $r = 13$ and $\tan θ = \frac{5}{12} .$

If $\tan θ = \frac{5}{12},$ and $\tan θ = \frac{y}{x},$ we first determine $r = \sqrt{x^{2} + y^{2}} = \sqrt{12^{2} + 5^{2}} = 13 .$ We then find $\cos θ = \frac{x}{r}$ and $\sin θ = \frac{y}{r} .$

\begin{array}{l} z = 13 (\cos θ + i \sin θ) \\ = 13 (\frac{12}{13} + \frac{5}{13} i) \\ = 12 + 5 i \end{array}

The rectangular form of the given number in complex form is $12 + 5 i .$

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Convert the complex number to rectangular form:

z = 4 (\cos \frac{11 π}{6} + i \sin \frac{11 π}{6})

$z = 2 \sqrt{3} - 2 i$

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Finding products of complex numbers in polar form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.

Kate Reply

To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.

Muhammed

What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?

Kate Reply

what is the change in momentum of a body?

Eunice Reply

what is a capacitor?

Raymond Reply

Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.

Gautam

A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate

Maria Reply

please solve

Sharon

8m/s²

Aishat

What is Thermodynamics

Muordit

velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.

Mehmet

A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat

Saheed Reply

50 m/s due south east

Someone

which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer

Ramon Reply

I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.

Someone

Scratch that

Someone

temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....

Someone

about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle

Someone

definitely of physics

Haryormhidey Reply

how many start and codon

Esrael Reply

what is field

Felix Reply

physics, biology and chemistry this is my Field

ALIYU

field is a region of space under the influence of some physical properties

Collete

what is ogarnic chemistry

WISDOM Reply

determine the slope giving that 3y+ 2x-14=0

WISDOM

Another formula for Acceleration

Belty Reply

a=v/t. a=f/m a

IHUMA

innocent

Adah

pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?

Nassze Reply

how do lnternal energy measures

Esrael

Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.

JALLAH Reply

No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.

Dlovan

Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?

Robert

like charges repel while unlike charges atttact

Raymond

What is specific heat capacity

Destiny Reply

Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).

AI-Robot

specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin

ROKEEB

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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