11. Polar Coordinate System Concept Overview

POLAR COORDINATE SYSTEM | CONCEPT OVERVIEW The topic of POLAR COORDINATE SYSTEMS can be referenced on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

CONCEPT INTRO: The Cartesian Coordinate System, is a RECTANGULAR COORDINATE SYSTEM, where “𝑥” and “𝑦” represent the directed distances from the coordinates axes to the point (𝑥, 𝑦).

The rectangular coordinates are a measure of how far away a point is from a reference axis: • The x-coordinate “𝑥” is the directed distance from the y-axis to the point of reference • The y-coordinate “𝑦” is the directed distance from the x-axis to the point of reference.



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In the POLAR COORDINATE SYSTEM, polar coordinates (𝑟, 𝜃) are determined by a distance (𝑟) and angle 𝜃 , that measures the angle between the positive 𝑥– 𝑎𝑥𝑖s and a ray that goes through the point. You can think of a polar coordinate system as how far away a coordinate is, and what it is relative to the origin.

To form the polar coordinate system in the plane, we determine the POLE or ORIGIN, and construct a RAY or line that is oriented relative to the POLAR AXIS. At the end of this line we find a point that has the coordinates (𝑟, 𝜃). We

can

convert

between

POLAR

COORDINATES

and

RECTANGULAR

COORDINATES by using known geometric relationships such as the Pythagorean theorem.



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The distance or ray “𝑟” represents the hypotenuse in the Pythagorean theorem, and helps us to relate the vertical and horizontal components as demonstrated in the figure above. The

FORMULAS

TO

CONVERT

BETWEEN

POLAR

AND

RECTANGULAR

COORDINATES can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

The “𝑥” and “𝑦” coordinates of the RECTANGULAR FORM can be determined from the radius 𝑟 and angle (𝜃) using the formulas: 𝑥 = 𝑟 cos 𝜃

𝑦 = 𝑟 sin 𝜃

tan 𝜃 =

4 5

Where: • 𝑟 is the distance from the origin to the point

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• 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray "𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠 The FORMULA FOR THE POLAR FORM A COMPLEX NUMBER can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The polar form of a nonzero complex number is represented by the expression: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 => Where: • 𝑟 is the modulus of the complex number • 𝜃 is the argument of the complex number

The radius and angle of the polar form of the nonzero complex number can be determined from the “𝑥” and “𝑦” coordinates using the formulas:

𝑟 = 𝑥 + 𝑗𝑦 = 𝜃 = arctan

𝑥? + 𝑦?

4 5

Where: • 𝑟 is the distance from the origin to the point

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• 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray “𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠 The FORMULAS AND VARIOUS FORMS OF EULER’S IDENTITY can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. EULER’S IDENTITY is used to express a complex number as related the trigonometric functions cosine and sine, which represents a power series of the function. 𝑒 => = cos 𝜃 + 𝑗 sin 𝜃 𝑒 A=> = cos 𝜃 − 𝑗 sin 𝜃

Using’s Euler’s Identity, we can rewrite exponential terms in terms of trigonometric functions, or to rewrite trigonometric functions in terms of exponential terms. 𝑒 => + 𝑒 A=> cos 𝜃 = 2 𝑒 => − 𝑒 A=> sin 𝜃 = 2𝑗



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CONCEPT EXAMPLE: The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

What is the rectangular form of the following polar equation: 𝑟 ? = 1 − tan? 𝜃 A. −𝑥 ? + 𝑥 D 𝑦 ? + 𝑦 ? = 0 B. 𝑥 ? + 𝑥 ? 𝑦 ? − 𝑦 ? + 𝑦 D = 0 C. −𝑥 D + 𝑦 ? = 0 D. 𝑥 D − 𝑥 ? + 𝑥 ? 𝑦 ? + 𝑦 ? = 0

SOLUTION: The

FORMULAS

TO

CONVERT

BETWEEN

POLAR

AND

RECTANGULAR

COORDINATES can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The radius and angle of the POLAR FORM can be determined from the “𝑥” and “𝑦” coordinates using the formulas:

𝑟 = 𝑥 + 𝑗𝑦 =

𝑥? + 𝑦? Made with

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𝜃 = arctan

4 5

Where: • 𝑟 is the distance from the origin to the point • 𝑥 is the x-coordiante of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray “𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠 Plugging in the values from the given complex expression, we find: 𝑟 ? = 1 − tan 𝜃 ? We can re-write the radius squared as a function of “𝑥” and “𝑦” coordinates to help us get both sides of the equation to have like terms:

𝑟 = 𝑥 + 𝑗𝑦 =

𝑥? + 𝑦?

?

𝑥? + 𝑦?

= 1 − tan? 𝜃

We can then replace the angle 𝜃 with the trigonometric expression that relates the angle to the “𝑥” and “𝑦” coordinates to help us get both sides of the equation to have like terms:

𝜃 = arctan 𝑥? + 𝑦?

4 5 ?

= 1 − tan? tanAF Made with

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Simplifying our expression, we find:

𝑥? + 𝑦? = 1 −

4G 5G

Therefore, the correct answer choice is D. 𝒙𝟐 + 𝒚𝟐 = 𝟏 −

𝒚𝟐 𝒙𝟐

MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM: The multiplication and division rules defined for complex numbers expressed in rectangular form can be applied to complex number expressed in polar form. The FORMULA TO CALCULATE THE PRODUCT OF TWO COMPLEX NUMBER IN POLAR REPRESENTATION can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.

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Using trigonometric identities, we calculate the product of complex numbers in polar form as: [𝑟F (cos 𝜃F + 𝑗 sin 𝜃F )][𝑟? (cos 𝜃? + 𝑗 sin 𝜃? )] = 𝑟F 𝑟? [cos 𝜃F + 𝜃? + 𝑗 sin(𝜃F + 𝜃? )] The FORMULA TO CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBER IN POLAR REPRESENTATION can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The quotient of complex numbers in polar form is expressed as: 𝑟F (cos 𝜃F + 𝑗 sin 𝜃F ) 𝑟F = cos 𝜃F − 𝜃? + 𝑗 sin(𝜃F − 𝜃? ) 𝑟? (cos 𝜃? + 𝑗 sin 𝜃? ) 𝑟?

Where: • 𝑟 is the absolute value of modulus of the complex number • 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray "𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠

CONCEPT EXAMPLE: The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

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What is the quotient of following complex numbers in polar representation? 𝑧F = 2(cos 120° + 𝑗 sin 120°) 𝑧? = 3(cos 30° + 𝑗 sin 30°) A.

?

B.

?

C.

?

D.

?

P P P P

cos 90° − 𝑗 sin 90° cos 90° + 𝑗 sin 90° sin 90° − 𝑗 cos 90° sin 90° + 𝑗 cos 90°

SOLUTION: We are given the two complex numbers in polar form: 𝑧F = 2(cos 120° + 𝑗 sin 120°) 𝑧? = 3(cos 30° + 𝑗 sin 30°) The FORMULA TO CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBER IN POLAR REPRESENTATION can be referenced under the topic of MATHEMATICS on



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page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We can calculate the quotient of two complex numbers in polar representation

using

the formula: 𝑟F (cos 𝜃F + 𝑗 sin 𝜃F ) 𝑟F = cos 𝜃F − 𝜃? + 𝑗 sin(𝜃F − 𝜃? ) 𝑟? (cos 𝜃? + 𝑗 sin 𝜃? ) 𝑟? Where: • 𝑟 is the absolute value of modulus of the complex number • 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray "𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠

Plugging in the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧F 2(cos 120° + 𝑗 sin 120°) = 𝑧? 3(cos 30° + 𝑗 sin 30°) We can simplify this expression using the formula to find the quotient of two complex numbers in polar representation:



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𝑟F (cos 𝜃F + 𝑗 sin 𝜃F ) 𝑟F = cos 𝜃F − 𝜃? + 𝑗 sin(𝜃F − 𝜃? ) 𝑟? (cos 𝜃? + 𝑗 sin 𝜃? ) 𝑟? Where: • 𝑟 is the absolute value of modulus of the complex number • 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray "𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠 We find the variable has the values of: • 𝑟F = 2 • 𝑟? = 3 • 𝜃F = 120° • 𝜃? = 30°

Plugging in the values from the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧F 2 = cos 120° − 30° + 𝑗 sin(120° − 30°) 𝑧? 3 Simplifying our expression, we find:



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𝑧F 2 = (cos 90° + 𝑗 sin 90°) 𝑧? 3 𝟐

Therefore, the correct answer choice is B. (𝐜𝐨𝐬 𝟗𝟎° + 𝐣 𝐬𝐢𝐧 𝟗𝟎°) 𝟑

DE MOIVRE’S FORMULA: The TOPIC OF DE MOIVRE’S FORMULA can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. DE MOIVRE’S THEOREM is used to calculate power and roots of complex numbers given in polar form. The FORMULA FOR DE MOIVRE’S THEOREM can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. DE MOIVRE’S FORMULA is used to relate complex numbers and trigonometry. We can use De Moivre’s Formula to find the powers of complex numbers in complex form. 𝑧 = 𝑥 + 𝑗𝑦

[

= [𝑟(cos 𝜃 + jsin 𝜃)][ = 𝑟 [ (cos 𝑛𝜃 + 𝑗 sin 𝑛𝜃)

Where: • 𝑛 is a positive integer



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We commonly use the following notation when working with complex number in polar form: 𝑟∠𝜃 = 𝑟 cos 𝜃 + 𝑗 sin 𝜃

[

In order to find the nth power of a complex number, we need to take the nth power of the absolute value or length and multiply the argument by “𝑟”. The FORMULA FOR THE ROOTS OF A COMPLEX NUMBER can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The kth root, “𝑤”, of a complex number is found from the equation expressed as: The “𝑘” roots of 𝑟(cos 𝜃 + 𝑗 sin 𝜃) can be found by substituting successively 𝑛 = 0,1,2, … (𝑘 − 1) in the formula:

𝑤=

b

𝑟 cos

𝜃 360° 𝜃 360° + 𝑛 + 𝑗 sin + 𝑛 𝑘 𝑘 𝑘 𝑘

If “𝑘” is any positive integer, any complex number (other than zero), has “𝑘” distinct roots. >

The amplitude of the first root, w is calculated as 𝑤F = . The point representing the d

roots 𝑤F , 𝑤? , … , 𝑤d are equal spaced on a unit circuits with a radius of the origin. The difference between each root on a unit circle can be calculated as:

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b

𝑟 drawn about

𝛥° =

360° 𝑘

CONCEPT EXAMPLE: The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

Find −1 + 3𝑗

F?

and write the result in standard form.

A. 3096 B. 4096 C. 5096 D. 6096

SOLUTION: The FORMULA FOR THE POLAR FORM A COMPLEX NUMBER can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The first step is to convert the complex expression to polar form to match the standard expression for polar form of nonzero complex number: 𝑧 = −1 + 3𝑗 The polar form of a nonzero complex number is represented by the expression:

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𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 => Where: • 𝑟 is the modulus of the complex number • 𝜃 is the argument of the complex number The radius and angle of the polar form of the nonzero complex number can be determined from the “𝑥” and “𝑦” coordinates using the formulas:

𝑟 = 𝑥 + 𝑗𝑦 = 𝜃 = arctan

𝑥? + 𝑦?

4 5

Where: • 𝑟 is the distance from the origin to the point • 𝑥 is the x-coordinate of the rectangular coordinates • 𝑦 is the y-coordinate of the rectangular coordinates • 𝜃 is the angle between the ray “𝑟” and the positive 𝑥 − 𝑎𝑥𝑖𝑠 Plugging in the values from the given expression, we can solve for modulus and argument of the complex number:

𝑟 = 𝑥 + 𝑗𝑦 =

𝑥? + 𝑦?

𝑟 = −1 + 𝑗 3 =



(−1)? +

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𝜃 = arctan

tan 𝜃 =

𝑦 𝑥

𝑦 3 = =− 3 𝑥 −1

𝜃 = arctan − 3 =

2𝜋 3

We then final polar form of the complex number can be re-written as: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 =>

𝑧 = −1 + 3𝑗 = 2 cos

2π 2𝜋 +𝑗 sin 3 3

The FORMULA FOR DE MOIVRE’S THEOREM can be referenced under the topic of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Using De Movire’s Theorem, we can expand the complex expression, such that we can then simplify it in the next step: 𝑧 = 𝑥 + 𝑗𝑦

[

= [𝑟(cos 𝜃 + jsin 𝜃)][ = 𝑟 [ (cos 𝑛𝜃 + 𝑗 sin 𝑛𝜃)

𝑧 = −1 + 3𝑗



F?

2𝜋 2𝜋 = 2 cos + 𝑗 sin 3 3

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𝑧 = 2F? cos

12 2𝜋 12 2𝜋 + 𝑗 sin 3 3

Simplifying we find the can simplifying the complex express to it’s final form: 𝑧 = 4096(cos 8𝜋 + 𝑗 sin 8𝜋) = 4096 1 + 𝑖 0

= 4096

Therefore, the correct answer choice is B. 𝟒𝟎𝟗𝟔



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