11.5 Polar Coordinate System Problem Set
POLAR COORDINATE SYSTEM | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.
PROBLEM 1: What are the polar coordinates of the point that is located at rectangular coordinates of (4,6)? A. (4, 6°) B. (4, 56.3°) C. (7.21, 33.7°) D. (7.21, 56.3°)
SOLUTION 1: The TOPIC OF POLAR COORDINATE SYSTEM 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 =
𝜃 = arctan
𝑥3 + 𝑦3
: ;
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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=
4
3
+ (6)3 = 7.211
𝜃 = arctan
A B
= 56.3°
Therefore, the correct answer choice is D. (𝟕. 𝟐𝟏, 𝟓𝟔. 𝟑°) PROBLEM 2: What is the polar form of the complex number 𝑧 = 3 + 4𝑖? A. (3)(cos 36.87° + 𝑖 sin 36.87°) B. (3)(cos 53.15° + 𝑖 sin 36.87°) C. (4)(cos 53.15° + 𝑖 sin 53.15°) D. (5)(cos 53.15° + 𝑖 sin 53.13°)
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SOLUTION 2: 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: 𝑥3 + 𝑦3
𝑟 = 𝑥 + 𝑗𝑦 =
𝜃 = arctan
: ;
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=
3
3
+ 4
𝜃 = arctan
B O
3
=5
= 53.13°
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Therefore, the correct answer choice is D. (𝟓)(𝐜𝐨𝐬 𝟓𝟑. 𝟏𝟑° + 𝒊 𝐬𝐢𝐧 𝟓𝟑. 𝟏𝟑°) PROBLEM 3: Given the polar coordinates of a point are (4, 120°), what are the rectangular coordinates? A. (2, 3.46) B. (3.46, 2) C. (−2, 3.46) D. (−2, −3.46)
SOLUTION 3: 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 radius and angle 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 =
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We can use the formula relating the tangent of the angle with the “𝑥” and “𝑦” coordinates to solve for the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒. 𝑦 = 𝑥 tan 120° 𝑦 = −1.7321𝑥 We can then use the formula relating the radius and rectangular coordinates to solve for the value of the x-coordinate.
𝑟 = 𝑥 + 𝑗𝑦 =
𝑥3 + 𝑦3
Plugging in the given radius value and 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 in terms of the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 we find: 4
3
= 𝑥 3 + −1.7321𝑥
3
Solving for the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 we find: 𝑥 = 4𝑥 3 = ±2 Since the point is in the second quadrant based on the angle, we know the x-value is negative. 𝑥 = −2
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Plugging this value of “𝑥” into our equation for the y-coordinate we find: 𝑦 = −1.7321 𝑎𝑛𝑑 𝑥= −1.7321 −2 = 3.46
Therefore, the correct answer choice is C. (−𝟐, 𝟑. 𝟒𝟔) PROBLEM 4: If the rectangular coordinates of a point are −3, −5.2 , what are the polar coordinates of the point? A. (−6, −120°) B. (6, −120°) C. (6, 120°) D. (6, −150°)
SOLUTION 4: The TOPIC OF POLAR COORDINATE SYSTEM 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 =
𝑥3 + 𝑦3
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𝑦 𝑥
𝜃 = arctan
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=
−3
𝜃 = arctan
3
+ −5.2 _`.3 _O
3
=6
= −120°
Therefore, the correct answer choice is B. (𝟔, 𝟏𝟐𝟎°) PROBLEM 5: What is the quotient of following complex numbers in polar representation? 𝑧b = 5(cos 330° + 𝑗 sin 330°) 𝑧3 = 9(cos 90° + 𝑗 sin 90°) `
A. (sin 240° + 𝑗 cos 240°) d
`
B. (sin 240° − 𝑗 cos 240°) d
`
C. (cos 240° + 𝑗 sin 240°) d
`
D. (cos 240° − 𝑗 sin 240°) d
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SOLUTION 5: We are given the two complex numbers in polar form: 𝑧b = 5(cos 330° + 𝑗 sin 330°) 𝑧3 = 9(cos 90° + 𝑗 sin 90°) 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. We can calculate the quotient of two complex numbers in polar representation the formula: 𝑟b (cos 𝜃b + 𝑗 sin 𝜃b ) 𝑟b = cos 𝜃b − 𝜃3 + 𝑗 sin(𝜃b − 𝜃3 ) 𝑟3 (cos 𝜃3 + 𝑗 sin 𝜃3 ) 𝑟3 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
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using
Plugging in the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧b 5(cos 330° + 𝑗 sin 330°) = 𝑧3 9(cos 90° + 𝑗 sin 90°) We can simplify this expression using the formula to find the quotient of two complex numbers in polar representation: 𝑟b (cos 𝜃b + 𝑗 sin 𝜃b ) 𝑟b = cos 𝜃b − 𝜃3 + 𝑗 sin(𝜃b − 𝜃3 ) 𝑟3 (cos 𝜃3 + 𝑗 sin 𝜃3 ) 𝑟3 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: • 𝑟b = 5 • 𝑟3 = 9 • 𝜃b = 330° • 𝜃3 = 90°
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Plugging in the values from the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧b 5 = cos 330° − 90° + 𝑗 sin(330° − 90°) 𝑧3 9 Simplifying our expression, we find: 𝑧b 5 = (cos 240° − 𝑗 sin 240°) 𝑧3 9 𝟓
Therefore, the correct answer choice is D. (𝐜𝐨𝐬 𝟐𝟒𝟎° − 𝐣 𝐬𝐢𝐧 𝟐𝟒𝟎°) 𝟗
PROBLEM 6: Using Euler’s Identity, rewrite the function below in exponential terms: sin3 𝑥 b
A. sin3 𝑥 = (2 cos 2𝑥 − 2) B
b
B. sin3 𝑥 = − (2 cos 2𝑥 + 2) B b
C. sin3 𝑥 = − (2 cos 2𝑥 − 2) B b
D. sin3 𝑥 = − (2 cos 2𝑥 + 2) B
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SOLUTION 6: 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. We are given a function that is written in trigonometric terms, and we would like to convert the expression to exponential terms using Euler’s Identity. sin3 𝑥 Using’s Euler’s Identity for sine, we can rewrite exponential terms in terms of trigonometric functions, or to rewrite trigonometric functions in terms of exponential terms.
sin 𝜃 =
𝑒 hi − 𝑒 _hi 2𝑗
The trick with this question is to realize that the sine term is squared, so we must also square the exponential side of the equation: 𝑒 hi − 𝑒 _hi sin 𝜃 = 2𝑗
3
3
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Simplifying the exponential expression and distributing the exponent, we re-write the exponential expression as: 1 h3; 𝑒 + 𝑒 _h3; − 2𝑒 j 4 1 sin3 𝑥 = − (2 cos 2𝑥 − 2) 4 sin3 𝑥 = −
𝟏
Therefore, the correct answer choice is C. 𝐬𝐢𝐧𝟐 𝐱 = − (𝟐 𝐜𝐨𝐬 𝟐𝐱 − 𝟐) 𝟒
PROBLEM 7: Find and write the result in standard form:
1 𝑗 3 − + 2 2
b
h O
3
3
b
h O
3
3
A. − − B. − + −
h O
D. +
b
h O
3
3
C.
b 3
bj
3
SOLUTION 7: 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.
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The first step is to convert the complex expression to polar form to match the standard expression for polar form of nonzero complex number: 1 𝑗 3 𝑧=− + 2 2 The polar form of a nonzero complex number is represented by the expression: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi 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
𝑥3 + 𝑦3
: ;
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
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Plugging in the values from the given expression, we can solve for modulus and argument of the complex number:
𝑟 = 𝑥 + 𝑗𝑦 =
𝑥3 + 𝑦3
1 𝑗 3 𝑟= − + = 2 2
𝜃 = arctan
𝜃 = arctan
1 − 2
3
3 2
+
3
=1
𝑦 𝑥
3 1 / − 2 2
=
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 hi 1 𝑗 3 2π 2𝜋 𝑧=− + = 1 cos +𝑗 sin 2 2 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.
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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
o
= [𝑟(cos 𝜃 + jsin 𝜃)]o = 𝑟 o (cos 𝑛𝜃 + 𝑗 sin 𝑛𝜃) bj
1 𝑗 3 𝑧= − + 2 2
𝑧 = 1bj cos
2𝜋 2𝜋 = 1 cos + 𝑗 sin 3 3
bj
10 2𝜋 10 2𝜋 + 𝑗 sin 3 3
Simplifying we find the can simplifying the complex express to it’s final form:
𝑧 = 1 cos
20𝜋 20𝜋 1 𝑗 3 + 𝑗 sin =− + 3 3 2 2 𝟏
𝐣 𝟑
𝟐
𝟐
Therefore, the correct answer choice is 𝐁. − + PROBLEM 8: Find the third root of the expression −2 − 2𝑗 3
A. 𝑧O =
t
B. 𝑧O =
t
C. 𝑧O =
t
D. 𝑧O =
t
4 − cos
bAu
4 −cos
bAu
d d
4 cos
bAu
4 cos
bAu
d d
− 𝑗 sin
bAu
+ 𝑗 sin
bAu
d d
− 𝑗 sin
bAu
+ 𝑗 sin
bAu
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SOLUTION 8: 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 = −2 − 2𝑗 3 The polar form of a nonzero complex number is represented by the expression: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi 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
𝑥3 + 𝑦3
: ;
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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 =
𝑥3 + 𝑦3
𝑟 = −2 − 2 3 =
3
−2
𝜃 = arctan
𝑦 𝑥
𝜃 = arctan
−2 3 4𝜋 = −2 3
+ −2 3
3
=4
We then final polar form of the complex number can be re-written as: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi
𝑧 = −2 − 2𝑗 3 = 4 cos
4π 4𝜋 +𝑗 sin 3 3
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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=
y
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. We will take our complex expression and set it equal to the equation for the roots of a complex number, with the calculate values plugged in:
t
−2 − 2𝑗 3 =
t
4𝜋 4𝜋 360° 360° 4 cos 3 + 𝑛 + 𝑗 sin 3 + 𝑛 𝑘 𝑘 𝑘 𝑘
To solve for the third root, we will plug in a value of 𝑘 = 3 𝑎𝑛𝑑 𝑛 = 2 into the equation above:
𝑧O =
t
4𝜋 4𝜋 360° 360° 4 cos 3 + (2) + 𝑗 sin 3 + (2) 3 3 3 3 Made with
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Simplifying we find the complex expression for the third root is:
𝑧O =
t
4 cos
4𝜋 4𝜋 4𝜋 4𝜋 + + 𝑗 sin + 9 3 9 3
𝑧O =
t
4 cos
16𝜋 16𝜋 + 𝑗 sin 9 9
Therefore, the correct answer choice is 𝐃. 𝐳𝟑 =
𝟑
𝟒 𝐜𝐨𝐬
𝟏𝟔𝛑 𝟏𝟔𝛑 + 𝐣 𝐬𝐢𝐧 𝟗 𝟗
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PROBLEM 1: What are the polar coordinates of the point that is located at rectangular coordinates of (4,6)? A. (4, 6°) B. (4, 56.3°) C. (7.21, 33.7°) D. (7.21, 56.3°)
SOLUTION 1: The TOPIC OF POLAR COORDINATE SYSTEM 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 =
𝜃 = arctan
𝑥3 + 𝑦3
: ;
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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=
4
3
+ (6)3 = 7.211
𝜃 = arctan
A B
= 56.3°
Therefore, the correct answer choice is D. (𝟕. 𝟐𝟏, 𝟓𝟔. 𝟑°) PROBLEM 2: What is the polar form of the complex number 𝑧 = 3 + 4𝑖? A. (3)(cos 36.87° + 𝑖 sin 36.87°) B. (3)(cos 53.15° + 𝑖 sin 36.87°) C. (4)(cos 53.15° + 𝑖 sin 53.15°) D. (5)(cos 53.15° + 𝑖 sin 53.13°)
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SOLUTION 2: 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: 𝑥3 + 𝑦3
𝑟 = 𝑥 + 𝑗𝑦 =
𝜃 = arctan
: ;
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=
3
3
+ 4
𝜃 = arctan
B O
3
=5
= 53.13°
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Therefore, the correct answer choice is D. (𝟓)(𝐜𝐨𝐬 𝟓𝟑. 𝟏𝟑° + 𝒊 𝐬𝐢𝐧 𝟓𝟑. 𝟏𝟑°) PROBLEM 3: Given the polar coordinates of a point are (4, 120°), what are the rectangular coordinates? A. (2, 3.46) B. (3.46, 2) C. (−2, 3.46) D. (−2, −3.46)
SOLUTION 3: 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 radius and angle 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 =
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We can use the formula relating the tangent of the angle with the “𝑥” and “𝑦” coordinates to solve for the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒. 𝑦 = 𝑥 tan 120° 𝑦 = −1.7321𝑥 We can then use the formula relating the radius and rectangular coordinates to solve for the value of the x-coordinate.
𝑟 = 𝑥 + 𝑗𝑦 =
𝑥3 + 𝑦3
Plugging in the given radius value and 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 in terms of the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 we find: 4
3
= 𝑥 3 + −1.7321𝑥
3
Solving for the 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 we find: 𝑥 = 4𝑥 3 = ±2 Since the point is in the second quadrant based on the angle, we know the x-value is negative. 𝑥 = −2
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Plugging this value of “𝑥” into our equation for the y-coordinate we find: 𝑦 = −1.7321 𝑎𝑛𝑑 𝑥= −1.7321 −2 = 3.46
Therefore, the correct answer choice is C. (−𝟐, 𝟑. 𝟒𝟔) PROBLEM 4: If the rectangular coordinates of a point are −3, −5.2 , what are the polar coordinates of the point? A. (−6, −120°) B. (6, −120°) C. (6, 120°) D. (6, −150°)
SOLUTION 4: The TOPIC OF POLAR COORDINATE SYSTEM 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 =
𝑥3 + 𝑦3
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𝑦 𝑥
𝜃 = arctan
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=
−3
𝜃 = arctan
3
+ −5.2 _`.3 _O
3
=6
= −120°
Therefore, the correct answer choice is B. (𝟔, 𝟏𝟐𝟎°) PROBLEM 5: What is the quotient of following complex numbers in polar representation? 𝑧b = 5(cos 330° + 𝑗 sin 330°) 𝑧3 = 9(cos 90° + 𝑗 sin 90°) `
A. (sin 240° + 𝑗 cos 240°) d
`
B. (sin 240° − 𝑗 cos 240°) d
`
C. (cos 240° + 𝑗 sin 240°) d
`
D. (cos 240° − 𝑗 sin 240°) d
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SOLUTION 5: We are given the two complex numbers in polar form: 𝑧b = 5(cos 330° + 𝑗 sin 330°) 𝑧3 = 9(cos 90° + 𝑗 sin 90°) 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. We can calculate the quotient of two complex numbers in polar representation the formula: 𝑟b (cos 𝜃b + 𝑗 sin 𝜃b ) 𝑟b = cos 𝜃b − 𝜃3 + 𝑗 sin(𝜃b − 𝜃3 ) 𝑟3 (cos 𝜃3 + 𝑗 sin 𝜃3 ) 𝑟3 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
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using
Plugging in the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧b 5(cos 330° + 𝑗 sin 330°) = 𝑧3 9(cos 90° + 𝑗 sin 90°) We can simplify this expression using the formula to find the quotient of two complex numbers in polar representation: 𝑟b (cos 𝜃b + 𝑗 sin 𝜃b ) 𝑟b = cos 𝜃b − 𝜃3 + 𝑗 sin(𝜃b − 𝜃3 ) 𝑟3 (cos 𝜃3 + 𝑗 sin 𝜃3 ) 𝑟3 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: • 𝑟b = 5 • 𝑟3 = 9 • 𝜃b = 330° • 𝜃3 = 90°
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Plugging in the values from the given complex expressions, we find the quotient of the two complex numbers is expressed as: 𝑧b 5 = cos 330° − 90° + 𝑗 sin(330° − 90°) 𝑧3 9 Simplifying our expression, we find: 𝑧b 5 = (cos 240° − 𝑗 sin 240°) 𝑧3 9 𝟓
Therefore, the correct answer choice is D. (𝐜𝐨𝐬 𝟐𝟒𝟎° − 𝐣 𝐬𝐢𝐧 𝟐𝟒𝟎°) 𝟗
PROBLEM 6: Using Euler’s Identity, rewrite the function below in exponential terms: sin3 𝑥 b
A. sin3 𝑥 = (2 cos 2𝑥 − 2) B
b
B. sin3 𝑥 = − (2 cos 2𝑥 + 2) B b
C. sin3 𝑥 = − (2 cos 2𝑥 − 2) B b
D. sin3 𝑥 = − (2 cos 2𝑥 + 2) B
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SOLUTION 6: 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. We are given a function that is written in trigonometric terms, and we would like to convert the expression to exponential terms using Euler’s Identity. sin3 𝑥 Using’s Euler’s Identity for sine, we can rewrite exponential terms in terms of trigonometric functions, or to rewrite trigonometric functions in terms of exponential terms.
sin 𝜃 =
𝑒 hi − 𝑒 _hi 2𝑗
The trick with this question is to realize that the sine term is squared, so we must also square the exponential side of the equation: 𝑒 hi − 𝑒 _hi sin 𝜃 = 2𝑗
3
3
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Simplifying the exponential expression and distributing the exponent, we re-write the exponential expression as: 1 h3; 𝑒 + 𝑒 _h3; − 2𝑒 j 4 1 sin3 𝑥 = − (2 cos 2𝑥 − 2) 4 sin3 𝑥 = −
𝟏
Therefore, the correct answer choice is C. 𝐬𝐢𝐧𝟐 𝐱 = − (𝟐 𝐜𝐨𝐬 𝟐𝐱 − 𝟐) 𝟒
PROBLEM 7: Find and write the result in standard form:
1 𝑗 3 − + 2 2
b
h O
3
3
b
h O
3
3
A. − − B. − + −
h O
D. +
b
h O
3
3
C.
b 3
bj
3
SOLUTION 7: 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.
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The first step is to convert the complex expression to polar form to match the standard expression for polar form of nonzero complex number: 1 𝑗 3 𝑧=− + 2 2 The polar form of a nonzero complex number is represented by the expression: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi 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
𝑥3 + 𝑦3
: ;
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
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Plugging in the values from the given expression, we can solve for modulus and argument of the complex number:
𝑟 = 𝑥 + 𝑗𝑦 =
𝑥3 + 𝑦3
1 𝑗 3 𝑟= − + = 2 2
𝜃 = arctan
𝜃 = arctan
1 − 2
3
3 2
+
3
=1
𝑦 𝑥
3 1 / − 2 2
=
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 hi 1 𝑗 3 2π 2𝜋 𝑧=− + = 1 cos +𝑗 sin 2 2 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.
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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
o
= [𝑟(cos 𝜃 + jsin 𝜃)]o = 𝑟 o (cos 𝑛𝜃 + 𝑗 sin 𝑛𝜃) bj
1 𝑗 3 𝑧= − + 2 2
𝑧 = 1bj cos
2𝜋 2𝜋 = 1 cos + 𝑗 sin 3 3
bj
10 2𝜋 10 2𝜋 + 𝑗 sin 3 3
Simplifying we find the can simplifying the complex express to it’s final form:
𝑧 = 1 cos
20𝜋 20𝜋 1 𝑗 3 + 𝑗 sin =− + 3 3 2 2 𝟏
𝐣 𝟑
𝟐
𝟐
Therefore, the correct answer choice is 𝐁. − + PROBLEM 8: Find the third root of the expression −2 − 2𝑗 3
A. 𝑧O =
t
B. 𝑧O =
t
C. 𝑧O =
t
D. 𝑧O =
t
4 − cos
bAu
4 −cos
bAu
d d
4 cos
bAu
4 cos
bAu
d d
− 𝑗 sin
bAu
+ 𝑗 sin
bAu
d d
− 𝑗 sin
bAu
+ 𝑗 sin
bAu
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SOLUTION 8: 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 = −2 − 2𝑗 3 The polar form of a nonzero complex number is represented by the expression: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi 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
𝑥3 + 𝑦3
: ;
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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 =
𝑥3 + 𝑦3
𝑟 = −2 − 2 3 =
3
−2
𝜃 = arctan
𝑦 𝑥
𝜃 = arctan
−2 3 4𝜋 = −2 3
+ −2 3
3
=4
We then final polar form of the complex number can be re-written as: 𝑧 = 𝑥 + 𝑗𝑦 = 𝑟(cos 𝜃 + 𝑗 sin 𝜃) = 𝑟𝑒 hi
𝑧 = −2 − 2𝑗 3 = 4 cos
4π 4𝜋 +𝑗 sin 3 3
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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=
y
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. We will take our complex expression and set it equal to the equation for the roots of a complex number, with the calculate values plugged in:
t
−2 − 2𝑗 3 =
t
4𝜋 4𝜋 360° 360° 4 cos 3 + 𝑛 + 𝑗 sin 3 + 𝑛 𝑘 𝑘 𝑘 𝑘
To solve for the third root, we will plug in a value of 𝑘 = 3 𝑎𝑛𝑑 𝑛 = 2 into the equation above:
𝑧O =
t
4𝜋 4𝜋 360° 360° 4 cos 3 + (2) + 𝑗 sin 3 + (2) 3 3 3 3 Made with
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Simplifying we find the complex expression for the third root is:
𝑧O =
t
4 cos
4𝜋 4𝜋 4𝜋 4𝜋 + + 𝑗 sin + 9 3 9 3
𝑧O =
t
4 cos
16𝜋 16𝜋 + 𝑗 sin 9 9
Therefore, the correct answer choice is 𝐃. 𝐳𝟑 =
𝟑
𝟒 𝐜𝐨𝐬
𝟏𝟔𝛑 𝟏𝟔𝛑 + 𝐣 𝐬𝐢𝐧 𝟗 𝟗
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