22.1 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 βπ₯β and βπ¦β represent the directed distances from the coordinates axes to the point (π₯, π¦).
The rectangular coordinates are a measure of how far away a point is from a reference axis: β’ The x-coordinate, denoted as βπ₯β, is the directed distance from the y-axis to the point of reference β’ The y-coordinate, denoted as βπ¦β, is the directed distance from the x-axis to the point of reference.
Made with
by Prepineer | Prepineer.com
In the POLAR COORDINATE SYSTEM, polar coordinates (π, π) are determined by the distance (π) and the angle π , which measures the angle between the positive π₯β ππ₯πs and a radius that goes through the point. You can think of a polar coordinate system as how far away a coordinate is, and how it is projected relative to the polar axis.
To form the polar coordinate system in the plane, we determine the POLE, otherwise known as an ORIGIN, and construct a RADIUS, or line, that is oriented relative to the POLAR AXIS. At the end of this RADIUS we define the point with the coordinates (π, π). We can convert between POLAR COORDINATES and RECTANGULAR COORDINATES by using known geometric relationships, such as the Pythagorean theorem.
Made with
by Prepineer | Prepineer.com
A generalized conversion schematic can be illustrated as:
The distance of the radius, βπβ, represents the hypotenuse in the Pythagorean theorem, and allows us to relate the vertical and horizontal components as demonstrated. The FORMULAS allow for us to convert BETWEEN POLAR AND RECTANGULAR COORDINATES can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The βπ₯β and βπ¦β coordinates in RECTANGULAR FORM can be determined from the radius π and angle (π) using the formulas: β’ π₯ = π cos π β’ π¦ = π sin π !
β’ tan π = ! Where:
β’ π is the distance from the origin to the point, the radius β’ π₯ is the x-coordinate of the rectangular coordinates
Made with
by Prepineer | Prepineer.com
β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ The FORMULA for the POLAR FORM A COMPLEX NUMBER can be referenced under the subject 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: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !" Where: β’ π is the modulus of the complex number β’ π is the argument of the complex number The radius and angle of a nonzero complex number in polar form can be determined from the βπ₯β and βπ¦β coordinates using the formulas: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordinate of the rectangular coordinates
Made with
by Prepineer | Prepineer.com
β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ The FORMULAS and VARIOUS FORMS OF EULERβS IDENTITY can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. EULERβS IDENTITY is used to express a complex number as it relates to the trigonometric functions cosine and sine, which represents a power series of the function. EULERβS IDENTITY states: π !" = cos π + π sin π π !!" = cos π β π sin π Usingβs EULERβS IDENTITY allows us to rewrite complex exponential expressions in terms of trigonometric functions, or vice versa, rewriting trigonometric functions in terms of exponential expressions: π !" + π !!" cos π = 2 π !" β π !!" sin π = 2π
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The rectangular form of the polar equation π ! = 1 β tan! π is best represented as: A. βπ₯ ! + π₯ ! π¦ ! + π¦ ! = 0 B. π₯ ! + π₯ ! π¦ ! β π¦ ! + π¦ ! = 0 C. βπ₯ ! + π¦ ! = 0 D. π₯ ! + π¦ ! = 1 β
!! !!
SOLUTION: The FORMULAS to CONVERT BETWEEN POLAR AND RECTANGULAR COORDINATES can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The radius and angle of an equation presented in POLAR FORM can be related to the βπ₯β and βπ¦β RECTANGULAR COORDINATES using the formulas: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Made with
by Prepineer | Prepineer.com
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordiante of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ We are given the following POLAR equation: π ! = 1 β tan π ! Using our established POLAR to RECTANGULAR relationships, we can re-write the radius squared as a function of βπ₯β and βπ¦β coordinates such that:
π = π₯ + ππ¦ =
π₯! + π¦!
So:
π₯! + π¦!
!
= 1 β tan! π
We can now aim to replace the angle π with the trigonometric expression that relates the angle to the βπ₯β and βπ¦β coordinates such that:
π = arctan
π¦ π₯
Made with
by Prepineer | Prepineer.com
Which further expands our original formula to read:
π₯!
+
π¦!
!
= 1 β tan
!
tan
!!
π¦ π₯
!
We now have our original equation in POLAR FORM represented as an equation in RECTANGULAR FORM. The next step would be to simplify this expression as much as possible. Doing so gives us: π¦! π₯ +π¦ =1β ! π₯ !
!
ππ
The correct answer choice is D. ππ + ππ = π β ππ
Made with
by Prepineer | Prepineer.com
MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM: The multiplication and division rules defined for complex numbers expressed in RECTANGULAR FORM can also be applied to complex numbers expressed in POLAR FORM. The GENERAL FORMULA to CALCULATE THE PRODUCT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Using trigonometric identities, we calculate the product of complex numbers in polar form using: [π! (cos π! + π sin π! )][π! (cos π! + π sin π! )] = π! π! [cos π! + π! + π sin(π! + π! )] The FORMULA to CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject 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: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π!
Made with
by Prepineer | Prepineer.com
Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The quotient of the following set of complex numbers in polar representation is: π§! = 2(cos 120Β° + π sin 120Β°) π§! = 3(cos 30Β° + π sin 30Β°) A.
!
B.
!
C.
!
! ! !
cos 90Β° β π sin 90Β° cos 90Β° + π sin 90Β° sin 90Β° β π cos 90Β°
!
D. ! sin 90Β° + π cos 90Β°
SOLUTION: We are given the two complex numbers in polar form, which are: β’ π§! = 2(cos 120Β° + π sin 120Β°) β’ π§! = 3(cos 30Β° + π sin 30Β°) We are asked to determine the quotient of these two numbers in POLAR FORM.
Made with
by Prepineer | Prepineer.com
The FORMULA to CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject of MATHEMATICS on 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 general formula: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π! Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ Plugging in the defined complex expressions, we find the quotient of the two complex numbers is expressed as: π§! 2(cos 120Β° + π sin 120Β°) = π§! 3(cos 30Β° + π sin 30Β°)
Made with
by Prepineer | Prepineer.com
We can simplify this expression using the general formula we are given for finding the quotient of two complex numbers in polar representation, such that: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π! Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ With our quotient defined, we can define each variable as: β’ π! = 2 β’ π! = 3 β’ π! = 120Β° β’ π! = 30Β° From here, we simply plug these variables in to our general formula, such that: π§! 2 = cos 120Β° β 30Β° + π sin(120Β° β 30Β°) π§! 3
Made with
by Prepineer | Prepineer.com
Simplifying our expression, we find that the quotient of the two complex numbers is most closely represented as: π§! 2 = (cos 90Β° + π sin 90Β°) π§! 3 π
The correct answer choice is B. (ππ¨π¬ ππΒ° + π£ π¬π’π§ ππΒ°) π
Made with
by Prepineer | Prepineer.com
DE MOIVREβS FORMULA: DE MOIVREβS THEOREM is used to calculate the power and roots of complex numbers through a relationship with trigonometric identities. The TOPIC of DE MOIVREβS FORMULA can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We can use De Moivreβs Formula to find the powers of complex numbers given in polar form. The general formula states: π§ = π₯ + ππ¦
!
= [π(cos π + jsin π)]! = π ! (cos ππ + π sin ππ)
Where: β’ π is a positive integer We commonly use the following notation when working with complex numbers in polar form: πβ π = π cos π + π sin π
!
Made with
by Prepineer | Prepineer.com
De Moivre's Theorem is useful is that it allows us to compute large powers of complex numbers expressed in polar form without having to carry out every multiplication explicitly. De Moivre's Theorem also allows us to establish compute nth roots of a complex number, where n is a positive integer. Now as if things havenβt already gotten deep, we are going to get even deeper, and more convolutedβ¦but just roll with us, make your way through attempting to understand the concepts the best you are able. We will be reinforcing through practice to make them more manageable. Letβs talk complex roots. The FORMULA for the ROOTS OF A COMPLEX NUMBER can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The kth root, βπ€β, of a complex number is found from the equation expressed as:
π€=
!
π cos
π 360Β° π 360Β° +π + π sin + π π π π π
The βπβ roots of π(cos π + π sin π) can be found by substituting successively π = 0,1,2, etc up to (π β 1) in the stated formula.
Made with
by Prepineer | Prepineer.com
If βπβ is any positive integer, any complex number (other than zero), has βπβ distinct roots. The amplitude of the first root, w, is calculated as:
π€! =
π π
The point representing the roots π€! , π€! , β¦ , π€! are equally spaced on a unit circle with a radius of
!
π drawn about the origin.
The difference between each root on a unit circle can be calculated as:
π₯Β° =
360Β° π
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
Find β1 + 3π
!"
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 subject 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 in to polar form to match the standard expression that we are accustomed to for the polar form of nonzero complex number. The complex number, in RECTANGULAR FORM, is given as: π§ = β1 + 3π
Made with
by Prepineer | Prepineer.com
The polar form of a nonzero complex number is represented by the expression: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !" Where: β’ π is the modulus of the complex number β’ π is the argument of the complex number The radius and angle in polar form of the nonzero complex number can be defined knowing the βπ₯β and βπ¦β coordinates of the complex number in RECTANGULAR FORM using: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ
Made with
by Prepineer | Prepineer.com
Taking these formulas and plugging in what we know, we can solve for the modulus and argument of the complex number in POLAR FORM, such that:
π = β1 + π 3 =
(β1)! +
3
!
=2
And:
π = arctan
3 2π = β1 3
The final polar form of the complex number can be re-written as: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !"
π§ = β1 + 3π = 2 cos
2Ο 2π +π sin 3 3
With the original complex expression now in POLAR FORM, we can move forward with defining the resultant. The FORMULA for DE MOIVREβS THEOREM can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. De Movireβs Theorem generally states: π§ = π₯ + ππ¦
!
= [π(cos π + jsin π)]! = π ! (cos ππ + π sin ππ) Made with by Prepineer | Prepineer.com
We know the POLAR FORM of our complex expression, which we defined in the previous step, and we also know that n is equal to 12. Therefore, using De Movireβs Theorem, we can expand the complex expression as:
π§ = β1 + 3π
!"
2π 2π = 2 cos + π sin 3 3
!"
Which gives us:
π§ = 2!" cos
12 2π 12 2π + π sin 3 3
And the standard form results as: π§ = 4096(cos 8π + π sin 8π) = 4096 1 + π 0
= 4096
The correct answer choice is B. ππππ
Made with
by Prepineer | Prepineer.com
CONCEPT INTRO: The Cartesian Coordinate System, is a RECTANGULAR COORDINATE SYSTEM, where βπ₯β and βπ¦β represent the directed distances from the coordinates axes to the point (π₯, π¦).
The rectangular coordinates are a measure of how far away a point is from a reference axis: β’ The x-coordinate, denoted as βπ₯β, is the directed distance from the y-axis to the point of reference β’ The y-coordinate, denoted as βπ¦β, is the directed distance from the x-axis to the point of reference.
Made with
by Prepineer | Prepineer.com
In the POLAR COORDINATE SYSTEM, polar coordinates (π, π) are determined by the distance (π) and the angle π , which measures the angle between the positive π₯β ππ₯πs and a radius that goes through the point. You can think of a polar coordinate system as how far away a coordinate is, and how it is projected relative to the polar axis.
To form the polar coordinate system in the plane, we determine the POLE, otherwise known as an ORIGIN, and construct a RADIUS, or line, that is oriented relative to the POLAR AXIS. At the end of this RADIUS we define the point with the coordinates (π, π). We can convert between POLAR COORDINATES and RECTANGULAR COORDINATES by using known geometric relationships, such as the Pythagorean theorem.
Made with
by Prepineer | Prepineer.com
A generalized conversion schematic can be illustrated as:
The distance of the radius, βπβ, represents the hypotenuse in the Pythagorean theorem, and allows us to relate the vertical and horizontal components as demonstrated. The FORMULAS allow for us to convert BETWEEN POLAR AND RECTANGULAR COORDINATES can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The βπ₯β and βπ¦β coordinates in RECTANGULAR FORM can be determined from the radius π and angle (π) using the formulas: β’ π₯ = π cos π β’ π¦ = π sin π !
β’ tan π = ! Where:
β’ π is the distance from the origin to the point, the radius β’ π₯ is the x-coordinate of the rectangular coordinates
Made with
by Prepineer | Prepineer.com
β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ The FORMULA for the POLAR FORM A COMPLEX NUMBER can be referenced under the subject 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: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !" Where: β’ π is the modulus of the complex number β’ π is the argument of the complex number The radius and angle of a nonzero complex number in polar form can be determined from the βπ₯β and βπ¦β coordinates using the formulas: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordinate of the rectangular coordinates
Made with
by Prepineer | Prepineer.com
β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ The FORMULAS and VARIOUS FORMS OF EULERβS IDENTITY can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. EULERβS IDENTITY is used to express a complex number as it relates to the trigonometric functions cosine and sine, which represents a power series of the function. EULERβS IDENTITY states: π !" = cos π + π sin π π !!" = cos π β π sin π Usingβs EULERβS IDENTITY allows us to rewrite complex exponential expressions in terms of trigonometric functions, or vice versa, rewriting trigonometric functions in terms of exponential expressions: π !" + π !!" cos π = 2 π !" β π !!" sin π = 2π
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The rectangular form of the polar equation π ! = 1 β tan! π is best represented as: A. βπ₯ ! + π₯ ! π¦ ! + π¦ ! = 0 B. π₯ ! + π₯ ! π¦ ! β π¦ ! + π¦ ! = 0 C. βπ₯ ! + π¦ ! = 0 D. π₯ ! + π¦ ! = 1 β
!! !!
SOLUTION: The FORMULAS to CONVERT BETWEEN POLAR AND RECTANGULAR COORDINATES can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The radius and angle of an equation presented in POLAR FORM can be related to the βπ₯β and βπ¦β RECTANGULAR COORDINATES using the formulas: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Made with
by Prepineer | Prepineer.com
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordiante of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ We are given the following POLAR equation: π ! = 1 β tan π ! Using our established POLAR to RECTANGULAR relationships, we can re-write the radius squared as a function of βπ₯β and βπ¦β coordinates such that:
π = π₯ + ππ¦ =
π₯! + π¦!
So:
π₯! + π¦!
!
= 1 β tan! π
We can now aim to replace the angle π with the trigonometric expression that relates the angle to the βπ₯β and βπ¦β coordinates such that:
π = arctan
π¦ π₯
Made with
by Prepineer | Prepineer.com
Which further expands our original formula to read:
π₯!
+
π¦!
!
= 1 β tan
!
tan
!!
π¦ π₯
!
We now have our original equation in POLAR FORM represented as an equation in RECTANGULAR FORM. The next step would be to simplify this expression as much as possible. Doing so gives us: π¦! π₯ +π¦ =1β ! π₯ !
!
ππ
The correct answer choice is D. ππ + ππ = π β ππ
Made with
by Prepineer | Prepineer.com
MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS IN POLAR FORM: The multiplication and division rules defined for complex numbers expressed in RECTANGULAR FORM can also be applied to complex numbers expressed in POLAR FORM. The GENERAL FORMULA to CALCULATE THE PRODUCT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Using trigonometric identities, we calculate the product of complex numbers in polar form using: [π! (cos π! + π sin π! )][π! (cos π! + π sin π! )] = π! π! [cos π! + π! + π sin(π! + π! )] The FORMULA to CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject 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: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π!
Made with
by Prepineer | Prepineer.com
Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The quotient of the following set of complex numbers in polar representation is: π§! = 2(cos 120Β° + π sin 120Β°) π§! = 3(cos 30Β° + π sin 30Β°) A.
!
B.
!
C.
!
! ! !
cos 90Β° β π sin 90Β° cos 90Β° + π sin 90Β° sin 90Β° β π cos 90Β°
!
D. ! sin 90Β° + π cos 90Β°
SOLUTION: We are given the two complex numbers in polar form, which are: β’ π§! = 2(cos 120Β° + π sin 120Β°) β’ π§! = 3(cos 30Β° + π sin 30Β°) We are asked to determine the quotient of these two numbers in POLAR FORM.
Made with
by Prepineer | Prepineer.com
The FORMULA to CALCULATE THE QUOTIENT OF TWO COMPLEX NUMBERS IN POLAR REPRESENTATION can be referenced under the subject of MATHEMATICS on 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 general formula: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π! Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ Plugging in the defined complex expressions, we find the quotient of the two complex numbers is expressed as: π§! 2(cos 120Β° + π sin 120Β°) = π§! 3(cos 30Β° + π sin 30Β°)
Made with
by Prepineer | Prepineer.com
We can simplify this expression using the general formula we are given for finding the quotient of two complex numbers in polar representation, such that: π! (cos π! + π sin π! ) π! = cos π! β π! + π sin(π! β π! ) π! (cos π! + π sin π! ) π! Where: β’ π is the absolute value of modulus of the complex number β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius "πβ and the positive π₯ β ππ₯ππ With our quotient defined, we can define each variable as: β’ π! = 2 β’ π! = 3 β’ π! = 120Β° β’ π! = 30Β° From here, we simply plug these variables in to our general formula, such that: π§! 2 = cos 120Β° β 30Β° + π sin(120Β° β 30Β°) π§! 3
Made with
by Prepineer | Prepineer.com
Simplifying our expression, we find that the quotient of the two complex numbers is most closely represented as: π§! 2 = (cos 90Β° + π sin 90Β°) π§! 3 π
The correct answer choice is B. (ππ¨π¬ ππΒ° + π£ π¬π’π§ ππΒ°) π
Made with
by Prepineer | Prepineer.com
DE MOIVREβS FORMULA: DE MOIVREβS THEOREM is used to calculate the power and roots of complex numbers through a relationship with trigonometric identities. The TOPIC of DE MOIVREβS FORMULA can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We can use De Moivreβs Formula to find the powers of complex numbers given in polar form. The general formula states: π§ = π₯ + ππ¦
!
= [π(cos π + jsin π)]! = π ! (cos ππ + π sin ππ)
Where: β’ π is a positive integer We commonly use the following notation when working with complex numbers in polar form: πβ π = π cos π + π sin π
!
Made with
by Prepineer | Prepineer.com
De Moivre's Theorem is useful is that it allows us to compute large powers of complex numbers expressed in polar form without having to carry out every multiplication explicitly. De Moivre's Theorem also allows us to establish compute nth roots of a complex number, where n is a positive integer. Now as if things havenβt already gotten deep, we are going to get even deeper, and more convolutedβ¦but just roll with us, make your way through attempting to understand the concepts the best you are able. We will be reinforcing through practice to make them more manageable. Letβs talk complex roots. The FORMULA for the ROOTS OF A COMPLEX NUMBER can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The kth root, βπ€β, of a complex number is found from the equation expressed as:
π€=
!
π cos
π 360Β° π 360Β° +π + π sin + π π π π π
The βπβ roots of π(cos π + π sin π) can be found by substituting successively π = 0,1,2, etc up to (π β 1) in the stated formula.
Made with
by Prepineer | Prepineer.com
If βπβ is any positive integer, any complex number (other than zero), has βπβ distinct roots. The amplitude of the first root, w, is calculated as:
π€! =
π π
The point representing the roots π€! , π€! , β¦ , π€! are equally spaced on a unit circle with a radius of
!
π drawn about the origin.
The difference between each root on a unit circle can be calculated as:
π₯Β° =
360Β° π
Made with
by Prepineer | Prepineer.com
POLAR COORDINATE SYSTEM | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
Find β1 + 3π
!"
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 subject 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 in to polar form to match the standard expression that we are accustomed to for the polar form of nonzero complex number. The complex number, in RECTANGULAR FORM, is given as: π§ = β1 + 3π
Made with
by Prepineer | Prepineer.com
The polar form of a nonzero complex number is represented by the expression: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !" Where: β’ π is the modulus of the complex number β’ π is the argument of the complex number The radius and angle in polar form of the nonzero complex number can be defined knowing the βπ₯β and βπ¦β coordinates of the complex number in RECTANGULAR FORM using: β’ π = π₯ + ππ¦ = β’ π = arctan
π₯! + π¦!
! !
Where: β’ π is the distance from the origin to the point β’ π₯ is the x-coordinate of the rectangular coordinates β’ π¦ is the y-coordinate of the rectangular coordinates β’ π is the angle between the radius βπβ and the positive π₯ β ππ₯ππ
Made with
by Prepineer | Prepineer.com
Taking these formulas and plugging in what we know, we can solve for the modulus and argument of the complex number in POLAR FORM, such that:
π = β1 + π 3 =
(β1)! +
3
!
=2
And:
π = arctan
3 2π = β1 3
The final polar form of the complex number can be re-written as: π§ = π₯ + ππ¦ = π(cos π + π sin π) = ππ !"
π§ = β1 + 3π = 2 cos
2Ο 2π +π sin 3 3
With the original complex expression now in POLAR FORM, we can move forward with defining the resultant. The FORMULA for DE MOIVREβS THEOREM can be referenced under the subject of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. De Movireβs Theorem generally states: π§ = π₯ + ππ¦
!
= [π(cos π + jsin π)]! = π ! (cos ππ + π sin ππ) Made with by Prepineer | Prepineer.com
We know the POLAR FORM of our complex expression, which we defined in the previous step, and we also know that n is equal to 12. Therefore, using De Movireβs Theorem, we can expand the complex expression as:
π§ = β1 + 3π
!"
2π 2π = 2 cos + π sin 3 3
!"
Which gives us:
π§ = 2!" cos
12 2π 12 2π + π sin 3 3
And the standard form results as: π§ = 4096(cos 8π + π sin 8π) = 4096 1 + π 0
= 4096
The correct answer choice is B. ππππ
Made with
by Prepineer | Prepineer.com
