9-5 Complex Numbers and De Moivre's Theorem

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1 Graph each number in the complex plane and find its absolute value 1 z = 4 + 4i 3 z = 4 6i For z = 4 6i, (a, b) = ( 4, 6) Graph the point ( 4, 6) in the complex plane For z = 4 + 4i, (a, b) = (4, 4) Graph the point (4, 4) in the complex plane Use the absolute value of a complex number formula Use the absolute value of a complex number formula 4 z = 5i z = 3 + i For z = 5i, (a, b) = (, 5) Graph the point (, 5) in the complex plane For z = 3 + i, (a, b) = ( 3, 1) Graph the point ( 3, 1) in the complex plane Use the absolute value of a complex number formula Use the absolute value of a complex number formula 5 z = 3 + 4i 3 z = 4 6i For z = 3 + 4i, (a, b) = (3, 4) Graph the point (3, 4) in the complex plane For z = 4 6i, (a, b) = ( 4, 6) Graph the point esolutions Manual - Powered by Cognero ( 4, 6) in the complex plane Page 1

2 5 z = 3 + 4i 7 z = 3 7i For z = 3 + 4i, (a, b) = (3, 4) Graph the point (3, 4) in the complex plane For z = 3 7i, (a, b) = ( 3, 7) Graph the point ( 3, 7) in the complex plane Use the absolute value of a complex number formula Use the absolute value of a complex number formula 6 z = 7 + 5i For z = 7 + 5i, (a, b) = ( 7, 5) Graph the point ( 7, 5) in the complex plane Use the absolute value of a complex number formula 7 z = 3 7i For z = 3 7i, (a, b) = ( 3, 7) Graph the point ( 3, 7) in the complex plane 8 z = 8 i For z = 8 i, (a, b) = (8, ) Graph the point (8, ) in the complex plane Use the absolute value of a complex number formula 9 VECTORS The force on an object given by z = i, where the components are measured in newtons (N) a Represent z as a vector in the complex plane b Find the magnitude and direction angle of the vector Page a For z = i, (a, b) = (10, 15) Graph the point (10, 15) in the complex plane Then draw a

3 9 VECTORS The force on an object given by z = The magnitude of the force about 1803 newtons at an angle of about 5631 Express each complex number in polar form i i, where the components are measured in newtons (N) a Represent z as a vector in the complex plane b Find the magnitude and direction angle of the vector 4 + 4i Find the modulus r and argument a For z = i, (a, b) = (10, 15) Graph the point (10, 15) in the complex plane Then draw a vector with an initial point at the origin and a terminal point at (10, 15) The polar form of 4+ 4i 11 + i + i Find the modulus r and argument b Use the absolute value of a complex number formula The polar form of + i 68) Find the measure of the angle that the vector makes with the positive real ax 1 4 (cos 68 + i sin i i 4 Find the modulus r and argument The magnitude of the force about 1803 newtons at an angle of about 5631 Express each complex number in polar form i 4 + 4i Find the modulus r and argument The polar form of 4 3 i (cos i sin 034) 13 i i Find the modulus r and argument The polar form of 4+ 4i Page 3

4 i The polar form of 4 3 (cos i sin 034) 13 i The polar form of + 4i 03) 16 1 (cos 03 + i sin i i Find the modulus r and argument i 1 Find the modulus r and argument The polar form of i The polar form of 1 i i i 4 + 5i Find the modulus r and argument The polar form of 4+ 5i 090) 3 + 3i Find the modulus r and argument (cos i sin The polar form of 3+ 3i i + 4i Find the modulus r and argument Graph each complex number on a polar grid Then express it in rectangular form 18 10(cos 6 + i sin 6) The value of r 10, and the value of the polar coordinates (10, 6) The polar form of + 4i 03) Plot (cos 03 + i sin i i 1 Find the modulus r and argument To express the number in rectangular form, evaluate the trigonometric values and simplify Page 4

5 9-5 Complex Numbers The polar form of 3+ 3i and De Moivre's Theorem Graph each complex number on a polar grid Then express it in rectangular form 18 10(cos 6 + i sin 6) The value of r, and the value of 3 Plot the polar coordinates (, 3) Notice that 3 radians slightly greater than but less than π 6 Plot To express the number in rectangular form, evaluate the trigonometric values and simplify The rectangular form of To express the number in rectangular form, evaluate the trigonometric values and simplify 19 (cos 3 + i sin 3) 19 (cos 3 + i sin 3) The value of r 10, and the value of the polar coordinates (10, 6) The rectangular form of The rectangular form of (cos 3 + i sin 3) i 0 The value of r, and the value of 3 Plot the polar coordinates (, 3) Notice that 3 radians slightly greater than but less than π The value of r 4, and the value of the polar coordinates To express the number in rectangular form, evaluate the trigonometric values and simplify The rectangular form of (cos 3 + i sin 3) i Plot To express the number in rectangular form, evaluate the trigonometric values and simplify Page 5

6 evaluate the trigonometric values and simplify The rectangular form of The rectangular form of (cos + i sin 3) 198Theorem 9-5 Complex Numbers and3de Moivre's + 08i 0 1 The value of r 4, and the value of the polar coordinates Plot The value of r 3, and the value of the polar coordinates Plot To express the number in rectangular form, evaluate the trigonometric values and simplify To express the number in rectangular form, evaluate the trigonometric values and simplify The rectangular form of The rectangular form of i 1 The value of r 3, and the value of the polar coordinates Plot The value of r 1, and the value of Plot the polar coordinates To express the number in rectangular form, Page 6

7 The rectangular form of The rectangular form of 9-5 Complex Numbers and De Moivre's Theorem i 3 The value of r 1, and the value of Plot the polar coordinates The value of r, and the value of the polar coordinates To express the number in rectangular form, evaluate the trigonometric values and simplify Plot To express the number in rectangular form, evaluate the trigonometric values and simplify The rectangular form of The rectangular form of 1 i 4 3(cos 180º + i sin 180º) 3 The value of r 3, and the value of Plot the polar coordinates ( 3, 180 ) The value of r, and the value of the polar coordinates 180 Plot To express the number in rectangular form, evaluate the trigonometric values and simplify Page 7 The rectangular form of 3(cos 180º + i sin 180º)

8 1 The rectangular form of The rectangular form of 3(cos 180º + i sin 180º) 3 i 4 3(cos 180º + i sin 180º) 5 (cos 360º + i sin 360º) The value of r 3, and the value of Plot the polar coordinates ( 3, 180 ) 180 The value of r, and the value of 360 Plot the polar coordinates To express the number in rectangular form, evaluate the trigonometric values and simplify To express the number in rectangular form, evaluate the trigonometric values and simplify The rectangular form of 3(cos 180º + i sin 180º) 3 5 (cos 360º + i sin 360º) The rectangular form of The value of r, and the value of (cos 360º + i sin 360º) 360 Plot the polar coordinates Find each product or quotient and express it in rectangular form 6 Use the Product Formula to find the product in polar form To express the number in rectangular form, evaluate the trigonometric values and simplify Now find the rectangular form of the product The rectangular of (cos esolutions Manual - Poweredform by Cognero 360º + i sin 360º) Page 8 The polar form The

9 The rectangular form of (cos 360º + i sin 360º) 9-5 Complex Numbers and De Moivre's Theorem Find each product or quotient and express it in rectangular form The polar form 10(cos i sin 180 ) The rectangular form Use the Quotient Formula to find the quotient in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form The polar form The polar form rectangular form The 7 5(cos i sin 135 ) (cos 45 + i sin 45 ) rectangular form The 9 (cos 90º + i sin 90º) (cos 70º + i sin 70º) Use the Product Formula to find the product in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form of the product The polar form 10(cos i sin 180 ) The rectangular form 10 The polar form rectangular form 4 The 30 8 Use the Quotient Formula to find the quotient in polar form Use the Quotient Formula to find the quotient in polar form Page 9

10 The polar form of the quotient 9-5 The polar form Complex Numbers rectangular form 4 The rectangular form of the quotient The and De Moivre's Theorem 30 3 (cos 60º + i sin 60º) 6(cos 150º + i sin 150º) Use the Quotient Formula to find the quotient in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form The polar form rectangular form The polar form of the quotient The The rectangular form 33 of the quotient Use the Quotient Formula to find the quotient in polar form 31 Use the Quotient Formula to find the quotient in polar form Now find the rectangular form Now find the rectangular form The polar form of the quotient The polar form of the quotient The rectangular form of the quotient 3 (cos 60º + i sin 60º) 6(cos 150º + i sin 150º) The rectangular form of the quotient 34 5(cos 180º + i sin 180º) (cos 135º + i sin 135º) Use the Product Formula to find the product inpage 10 polar form

11 The rectangular form of the The polar form of the quotient The rectangular form of the quotient 34 5(cos 180º + i sin 180º) (cos 135º + i sin 135º) Use the Product Formula to find the product in polar form quotient Find each power and express it in rectangular form 36 ( + i)6 First, write + i in polar form Now find the rectangular form of the product The polar form The rectangular form The polar form of + i Now use De Moivre s Theorem to find the sixth power 35 Use the Quotient Formula to find the quotient in polar form Therefore, 37 (1i 5)3 Now find the rectangular form of the product First, write 1i 5 in polar form The polar form of the quotient The rectangular form of the quotient The polar form of 1i 5 Now use De Moivre s Theorem to find the third power Find each power and express it in rectangular form 36 ( + i)6 First, write + i in polar form Therefore, Page 11 38

12 9-5 Complex Therefore, Numbers and De Moivre's Theorem 37 (1i 5)3 39 ( i) 3 First, write 1i 5 in polar form First, write i in polar form The polar form of 1i 5 Now use De Moivre s Theorem to find the third power The polar form of i Now use De Moivre s Theorem to find the third power Therefore, 38 already in polar form Use De Moivre s Theorem to find the fourth power Therefore, 40 (3 5i)4 First, write 3 5i in polar form 39 ( i) 3 First, write i in polar form The polar form of 3 5i Now use De Moivre s Theorem to find the fourth power The polar form of i Now use De Moivre s Theorem to find the third power Therefore, Page 1 41 ( + 4i) 4

13 9-5 Complex De Moivre's Theorem Therefore, Numbers and Therefore, 4 (3 6i)4 40 (3 5i)4 First, write 3 5i in polar form First, write 3 6i in polar form The polar form of 3 5i The polar form of 3 6i Now use De Moivre s Theorem to find the fourth power Now use De Moivre s Theorem to find the fourth power Therefore, Therefore, 43 ( + 3i) 41 ( + 4i)4 First, write + 3i in polar form First, write + 4i in polar form The polar form of + 3i Now use De Moivre s Theorem to find the second power The polar form of + 4i Now use De Moivre s Theorem to find the fourth power Therefore, Therefore, 4 (3 6i)4 First, write 3 6i in polar form 44 already in polar form Use De Moiver s Theorem to find the third power Page 13

14 6 Therefore, 6 The equation x 1 = 0 can be written as x = 1 To find the location of the six vertices for one of the hexagons, find the sixth roots of 1 First, write 1 in polar form 44 already in polar form Use De Moiver s Theorem to find the third power The polar form of 1 1 (cos 0 + i sin 0) Now write an expression for the sixth roots 45 Let n = 0 to find the first root of 1 already in polar form Use De Moiver s Theorem to find the fourth power Notice that the modulus of each complex number 1 The arguments are found by, resulting in 46 DESIGN Stella works for an adverting agency She wants to incorporate a design compred of regular hexagons as the artwork for one of her proposals Stella can locate the vertices of one of the central regular hexagons by graphing the 6 solutions to x 1 = 0 in the complex plane Find the vertices of th hexagon increasing by for each successive root Therefore, we can calculate the remaining roots by adding to each previous n =1 n = 6 6 The equation x 1 = 0 can be written as x = 1 To find the location of the six vertices for one of the hexagons, find the sixth roots of 1 First, write 1 in polar form n =3 n =4 n =5 Page 14

15 n =4 Let n = n =5 The vertices are located at Let n = 3 Find all of the dtinct p th roots of the complex number 47 sixth roots of i First, write i in polar form The polar form of i 1(cos Let n = 4 + i sin ) Now Let n = 4 write an expression for the sixth roots Let n = 0, 1,, 3, 4 and 5 successively to find the sixth roots Let n = 0 The sixth roots of i are approximately i, i, i, i, i, i 48 fifth roots of i First, write i in polar form Let n = 1 The polar form of i Now write an expression for the fifth roots Let n = Page 15 Let n = 0, 1,, 3 and 4 successively to find the fifth

16 The sixth roots of i are approximately i, i, i, i, i 49 fourth roots of 4 The polar form of i and De Moivre's Theorem 9-5 Complex Numbers Now write an expression for the fifth roots 4i First, write 4 4i in polar form Let n = 0, 1,, 3 and 4 successively to find the fifth roots Let n = 0 The polar form of 4 4i Now write an expression for the fourth roots Let n = 1 Let n = 0, 1, and 3 successively to find the fourth roots Let n = Let n = 0 Let n = 3 Let n = 1 Let n = 4 Let n = The sixth roots of i are approximately i, i, i, i, i Let n = 3 49 fourth roots of 4 4i First, write 4 4i in polar form Page 16 The fourth roots of 4 4i are approximately i , i, 0 167i, 167

17 Let n = 3 The cube roots of i are approximately 3 + 4i, i, i 51 fifth roots of The fourth roots of 4 4i are approximately i, i, 0 167i, 167 0i i First, write i in polar form 50 cube roots of i First, write i in polar form i The polar form of (cos i sin 163) Now write an expression for the fifth roots The polar form of i 15(cos 78 + i sin 78) Now write an expression for the cube roots Let n = 0, 1,, 3, and 4 successively to find the fifth roots Let n = 0, 1 and successively to find the cube roots Let n = 0 Let n = 0 Let n = 1 Let n = 1 Let n = Let n = Let n = 3 The cube roots of i are approximately 3 + 4i, i, i 51 fifth roots of i esolutions Manual - Powered by Cognero First, write i in polar form Let n = 4 Page 17

18 The square roots of 3 4i are approximately 1 + i and 1 i Let n = 4 53 find the square roots of unity First, write 1 in polar form i are approximately The fifth roots of i i , , i, i, and i 5 square root of 3 4i First, write 3 4i in polar form The polar form of 1 1 (cos 0 + i sin 0) Now write an expression for the square roots Let n = 0 to find the first root of 1 The polar form of 3 4i 5(cos i sin 407) Now write an expression for the square roots Let n = 0 and 1 successively to find the square roots Let n = 0 Notice that the modulus of each complex number 1 The arguments are found by nπ, resulting in increasing by nπ for each successive root Therefore, we can calculate the remaining root by adding nπ to the previous n =1 The square roots of 1 are ±1 54 find the fourth roots of unity First, write 1 in polar form Let n = 1 The square roots of 3 4i are approximately 1 + i and 1 i The polar form of 1 1 (cos 0 + i sin 0) Now write an expression for the fourth roots 53 find the square roots of unity First, write 1 in polar form Let n = 0 to find the first root of 1 Page 18

19 adding nπ to the previous n =1 n =3 The square roots of 1 are ±1 54 find the fourth roots of unity First, write 1 in polar form The fourth roots of 1 are ±1 and ±i 55 ELECTRICITY The impedance in one part of a series circuit 5(cos 09 + j sin 09) ohms In the second part of the circuit, it 8(cos 04 + j sin 04) ohms a Convert each expression to rectangular form b Add your answers from part a to find the total impedance in the circuit c Convert the total impedance back to polar form The polar form of 1 1 (cos 0 + i sin 0) Now write an expression for the fourth roots Let n = 0 to find the first root of 1 a Evaluate the trigonometric values and simplify b Find the sum ( j ) + ( j ) = j ohms c Find the modulus r and argument Notice that the modulus of each complex number 1 The arguments are found by, resulting in increasing by for each successive root Therefore, we can calculate the remaining roots by adding to each previous n =1 n = n =3 The polar form of j 163(cos j sin 059) ohms Find each product Then repeat the process by multiplying the polar forms of each pair of complex numbers using the Product Formula 56 (1 i)(4 + 4i) First, find each product The fourth roots of 1 are ±1 and ±i 55 ELECTRICITY The impedance in one part of a series circuit 5(cos 09 + j sin 09) ohms In the second part of the circuit, it 8(cos 04 + j sin 04) ohms a Convert each expression to rectangular form b Add your answers from part a to find the total impedance in the circuit c Convert the total impedance back to polar form Express each complex number in polar form For 1 i, find the modulus r and argument Page 19 a Evaluate the trigonometric values and simplify The polar form of 1 i

20 Now find the rectangular form of the product The polar form of and 704j De 163(cos 059 Theorem Complex Numbers Moivre's j sin 059) ohms Find each product Then repeat the process by multiplying the polar forms of each pair of complex numbers using the Product Formula 56 (1 i)(4 + 4i) 57 (3 + i)(3 i) First, find each product First, find each product Express each complex number in polar form For 3 + i, find the modulus r and argument Express each complex number in polar form For 1 i, find the modulus r and argument The polar form of 3 + i For 3 i, find the modulus r and argument The polar form of 1 i For 4 + 4i, find the modulus r and argument The polar form of 3 i The polar form of 4 + 4i Use the Product Formula to find the product in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form of the product 58 (4 + i)(3 i) First, find each product 57 (3 + i)(3 i) First, find each product Express each complex number in polar form Express each complex number in polar form For 3 i, find the modulus r and argument Page 0

21 Now find the rectangular form of the product Now find the rectangular form of the product 58 (4 + i)(3 i) 59 ( 6 + 5i)( 3i) First, find each product First, find each product Express each complex number in polar form Express each complex number in polar form For 3 i, find the modulus r and argument For 6 + 5i, find the modulus r and argument The polar form of 3 i The polar form of 6 + 5i For 4 + i, find the modulus r and argument For 3i, find the modulus r and argument The polar form of 4 + i The polar form of 3i Use the Product Formula to find the product in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form of the product 59 ( 6 + 5i)( 3i) 60 ( + i)(1 + i) First, find each product First, find each product Express each complex number in polar form For 6 + 5i, find the modulus r and argument Page 1 Express each complex number in polar form

22 Now find the rectangular form of the product Now find the rectangular form of the product 60 ( 61 (3 i)(1 + + i)(1 + i) i) First, find each product First, find each product Express each complex number in polar form For + i, find the modulus r and argument Express each complex number in polar form For 3 i, find the modulus r and argument The polar form of 3 i The polar form of + i For 1 + i, find the modulus r and argument For 1 + i, find the modulus r and argument The polar form of 1 + The polar form of 1 + i i Use the Product Formula to find the product in polar form Use the Product Formula to find the product in polar form Now find the rectangular form of the product Now find the rectangular form of the product 61 (3 i)(1 + i) First, find each product 6 FRACTALS A fractal a geometric figure that made up of a pattern that repeated indefinitely on successively smaller scales, as shown below Refer to the image on Page 578 In th problem, you will generate a fractal through iterations of f (z) = z Consider z 0 = iPage a Calculate z 1, z, z 3, z 4, z 5, z 6, and z 7 where z 1 = f (z 0), z = f (z 1), and so on

23 6 FRACTALS A fractal a geometric figure that made up of a pattern that repeated indefinitely on successively smaller scales, as shown below 9-5 Complex Numbers and De Moivre's Theorem Refer to the image on Page 578 In th problem, you will generate a fractal through iterations of f (z) = z Consider z 0 = i Use the expression that you found for z 4 to find z 5 a Calculate z 1, z, z 3, z 4, z 5, z 6, and z 7 where z 1 = f (z 0), z = f (z 1), and so on b Graph each of the numbers on the complex plane c Predict the location of z 100 Explain Use the expression that you found for z 5 to find z 6 a Calculate z 1 Use the expression that you found for z 1 to find z Use the expression that you found for z 6 to find z 7 Use the expression that you found for z to find z 3 b For z 1 = i, (a, b) = (039, 08) For z i, (a, b) = ( 049, 06) For z i, (a, b) = ( 014, 061) For z i, (a, b) = ( 035, 017) For z i, (a, b) = (009, 01) For z i, (a, b) = ( 0006, Use the expression that you found for z 3 to find z 4 00) For z i, (a, b) = ( 00004, 00003) Graph the points in the complex plane Use the expression that you found for z 4 to find z 5 Page c As more iterates are calculated and graphed, the3 iterates approach the origin Sample answer: z 100 will be located very close to the origin With each

24 same, but the imaginary component changed signs Therefore, the transformation applied to point z to obtain point w a reflection in the real ax Let z = (1, 6i) w = (1, 6i) c As more iterates are calculated and graphed, the iterates approach the origin Sample answer: z 100 will be located very close to the origin With each iteration of f (z) = z, the iterates approach the origin 63 TRANSFORMATIONS There are certain operations with complex numbers that correspond to geometric transformations in the complex plane Describe the transformation applied to point z to obtain point w in the complex plane for each of the following operations a w = z + (3 4i) b w the complex conjugate of z c w = i z d w = 05z c Let z = a + b i For z, z = (a, b) Use substitution to find w a Let z = a + b i For z, z = (a, b) For the transformation t represented by 3 4i, t = (3, 4) So, w = (a, b) + (3, 4) or (a + 3, b 4) Thus, the transformation applied to point z to obtain point w a translation 3 units to the right and 4 units down Let z = (1, 6i) w = (4, i) So, w = b + a i Thus, w = ( b, a) Therefore, the transformation applied to point z to obtain point w a rotation of 90 counterclockwe about the origin Let z = (1, 6i) w = ( 6, i) b Let z = a + b i For z, z = (a, b) Let w be the complex conjugate of z So, w = a b i Thus, w = (a, b) Notice that the real component stayed the same, but the imaginary component changed signs Therefore, the transformation applied to point z to obtain point w a reflection in the real ax Let z = (1, 6i) w = (1, 6i) d Let z = a + b i For z, z = (a, b) Use substitution to find w So, w = 05a + 05b i Thus, w = (05a, 05b) Page 4 Therefore, the transformation applied to point z to obtain point w a dilation by a factor of 05

25 to find w z = 15 To find the cube roots of 15, write 15 in polar form So, w = 05a + 05b i Thus, w = (05a, 05b) Therefore, the transformation applied to point z to obtain point w a dilation by a factor of 05 Let z = (1, 6i) w = (05, 15i) The polar form of 15 15(cos π + i sin π) Now write an expression for the cube roots Let n = 0, 1, and successively to find the cube roots Let n = 0 Find z and the p th roots of z given each of the following 64 p = 3, one cube root Cube to find z Let n = 1 Let n = z = 15 To find the cube roots of 15, write 15 in polar form z = 15 and the cube roots of 15 are 5,and, Page 5

26 Let n = 0, 1,, and 3 successively to find the cube roots Let n = 0 z = 15 and the cube roots of 15 are 5,and, 65 p = 4, one fourth root 1 i First, write 1 i in polar form Let n = 1 The polar form of 1 i Let n = Use De Moivre s Theorem to find z To find the fourth roots of 4, write 4 in polar form Let n = 3 The polar form of 4 4(cos π + i sin π) Now write an expression for the cube roots z = 4 and the fourth roots of 4 are 1 + i, 1 + i, 1 i, and 1 i 66 GRAPHICS By representing each vertex by a Let n = 0, 1,, and 3 successively to find the cube roots complex number in polar form, a programmer dilates and then rotates the square below 45 counterclockwe so that the new vertices lie at the midpoints of the sides of the original square Let n = 0 Page 6

27 66 GRAPHICS By representing each vertex by a 9-5 complex number in polar form, a programmer The polar form of (, 0) dilates and then rotates the square below 45 Complex Numbers and De Moivre's Theorem substitution to solve for z counterclockwe so that the new vertices lie at the midpoints of the sides of the original square Use To express z in rectangular form, evaluate the trigonometric values and simplify a By what complex number should the programmer multiply each number to produce th transformation? b What happens if the numbers representing the original vertices are multiplied by the square of your answer to part a? a When the complex number representing the vertex (, ) multiplied by the complex number z, the product will be located at the point (, 0) because (, 0) the midpoint of the side of the original square located 45 counterclockwe of (, ) Write the vertex (, ) in polar form The programmer should multiply by b Square the answer found in part a The vertex (, ) can be written as + i Multiply The polar form of (, ) th vertex by i Write (, 0) in polar form For 1 + i, (a, b) = ( 1, 1) The vertex (, ) can be written as + i The polar form of (, 0) Use Multiply th vertex by i substitution to solve for z For 1 i, (a, b) = ( 1, 1) The vertex (, ) can be written as ipage 7 Multiply th vertex by i

28 9-5 Complex Numbers and De Moivre's Theorem For 1 i, (a, b) = ( 1, 1) The vertex (, ) can be written as i Let n = 0, 1, and successively to find the cube roots i Multiply th vertex by Let n = 0 For 1 i, (a, b) = (1, 1) The vertex (, ) can be written as i Multiply th vertex by Let n = 1 i For 1 + i, (a, b) = (1, 1) The vertices of the square are being rotated 90 counterclockwe and are dilated by a factor of Let n = Use the Dtinct Roots Formula to find all of the solutions of each equation Express the solutions in rectangular form 67 x3 = i The cube roots of i are Solve for x, i Thus, the solutions to the equation are Find the cube roots of i First, write i in polar form,and,,and i 68 x3 + 3 = 18 Solve for x The polar form of i cos + i sin Now write an expression for the cube roots LetManual n = 0,- Powered 1, and bysuccessively esolutions Cognero roots to find the cube Find the cube roots of 15 First, write 15 in polar form Page 8 The polar form of cos 0 + i sin 0 Now write an expression for the cube roots

29 The cube roots of 15 are 5,,and Find the cube roots of 15 First, write 15 in polar form Thus, the solutions to the equation are 5,,and 69 x4 = 81i The polar form of cos 0 + i sin 0 Now write an expression for the cube roots Solve for x Find the fourth roots of 81i First, write 81i in polar form Let n = 0, 1, and successively to find the cube roots Let n = 0 The polar form of 81i Now write an expression for the fourth roots Let n = 1 Let n = 0, 1,, and 3 successively to find the fourth roots Let n = 0 Let n = Let n = 1 The cube roots of 15 are 5,,and Let n = Thus, the solutions to the equation are 5,,and 69 x4 = 81i Page 9

30 Let n = Let n = 1 Let n = 3 Let n = Thus, the solutions to the equation are i, i, i, and i Let n = 3 70 x5 1 = 103 Solve for x Let n = 4 Find the fifth roots of 104 First, write 104 in polar form Thus, the solutions to the equation are 4, i, i, 34 35i, and i The polar form of (cos 0 + i sin 0) Now write an expression for the fourth roots 71 x3 + 1 = i Solve for x Let n = 0, 1,, 3, and 4 successively to find the fifth roots Find the cube roots of 1 + i First, write 1 + i in polar form Let n = 0 Let n = 1 Page 30 The polar form of 1 + i

31 i, and i 7 x4 + i = 1 Find the cube roots of 1 + i First, write 1 + i in Solve for x polar form Find the fourth roots of 1 i First, write 1 i in polar form The polar form of 1 + i Now write an expression for the cube roots The polar form of 1 i Now write an expression for the fourth roots Let n = 0, 1, and successively to find the cube roots Let n = 0 Let n = 0, 1,, and 3 successively to find the fourth roots Let n = 0 Let n = 1 Let n = 1 Let n = Let n = Thus, the solutions to the equation are i, i, and i 7 x4 + i = 1 Solve for x Find the fourth roots of 1 i First, write 1 i in polar form Let n = 3 Page 31

32 Thus, the solutions to the equation are i, i, i, and i Let n = 3 73 ERROR ANALYSIS Alma and Blake are evaluating Alma uses DeMoivre s + i sin Theorem and gets an answer of cos Thus, the solutions to the equation are i, i, i, and i 73 ERROR ANALYSIS Alma and Blake are evaluating part of the problem Is either of them correct? Explain your reasoning Alma uses DeMoivre s To evaluate + i sin Theorem and gets an answer of cos Blake tells her that she has only completed in, first write polar form Blake tells her that she has only completed part of the problem Is either of them correct? Explain your reasoning To evaluate, first write in polar form The polar form of Now use DeMoivre s Theorem to find the fifth power The polar form of Now use DeMoivre s Theorem to find the fifth power Therefore, So, Blake correct Sample answer: Alma only converted the expression into polar form She needed to use DeMoivre s Theorem to find the fifth power 74 REASONING Suppose z = a + b i one of the Therefore, So, Blake 9th roots of 1 a What the maximum value of a? b What the maximum value of b? Page 3

33 correct Sample answer: Alma only converted the expression into polar form She needed to use From the table, it appears that y achieves a DeMoivre s Theorem to find the fifth power maximum value of about when x = 7 Thus, when n = 7, the expression sin will achieve a 74 REASONING Suppose z = a + b i one of the 9th roots of 1 maximum value of sin or about a What the maximum value of a? b What the maximum value of b? CHALLENGE Find the roots shown on each graph and write them in polar form Then identify the complex number with the given To find the 9th roots of unity, first write 1 in polar roots form The polar form of 1 1 (cos 0 + i sin 0) Now write an expression for the 9th roots 75 Let r be any of the roots depicted Since r lies on a circle of radius 3, The root r1 at can be represented by at The value of a represented by the expression cos Th expression evaluated for integer can be represented by can The root r3 at values of n from 0 to 8 The range of the cosine function 1 y 1 So, the greatest value that th expression can achieve 1 Thus, the maximum value of a 1 be represented by To determine the number whose roots are r1, r, and b The expression for the 9th roots of 1 r3, use De Moivre s Theorem to cube any one of them The value of b represented by the expression sin The root r Th expression evaluated for integer values of n from 0 to 8 Use a graphing calculator to find the maximum value of sin Enter sin in the Y= menu Use the TABLE function to view the values of y for the different integer values of x The roots in polar form are The complex number with the given roots 7i From the table, it appears that y achieves a maximum value of about when x = 7 Thus, when n = 7, the expression sin will achieve a maximum value of sin or about CHALLENGE Find the roots shown on each graph and write them in polar form Then Page 33 76

34 The roots in polar form are The roots in polar form are The The 9-5 Complex Numbers Theorem complex number with the and given De rootsmoivre's 7i complex number with the given roots PROOF Given z 1 = r1(cos = r(cos i sin 1) and z ), where r 0, prove that + i sin [cos ( 1 = 1+ ) + i sin ( 1 )] 76 Given: z 1 = r1 (cos (cos Let r be any of the roots depicted Since r lies on a circle of radius, The root r1 at can be + i sin Prove: represented by The root r 1 + i sin 1) and z = r ) [cos ( 1 = ) + i sin ( 1 )] can be represented by at = can The root r3 at be represented by The can be represented by root r4 at To determine the = number whose roots are r1, r, r3, and r4, use De Moivre s Theorem to rae any one of them to the fourth power (cos 1 = cos = The roots in polar form are The complex number with the given roots PROOF Given z 1 = r1(cos = r(cos = + i sin [cos ( i sin 1) and z ), where r 0, prove that ) + i sin ( 1 )] Given: z 1 = r1 (cos (cos + i sin 1 + i sin 1) ) - Powered by Cognero esolutions Manual Prove: and z = r cos + sin [(cos cos 1 1 sin 1 sin cos cos [cos ( 1 ) cos 1 + i sin ) + sin i sin 1 sin ) + i sin ( 1 )] + i(sin 1 1)] REASONING Determine whether each statement sometimes, always, or never true Explain your reasoning 78 The p th roots of a complex number z are equally spaced around the circle centered at the origin with radius The expression for the dtinct roots of a complex number or Page 34 Each root will have the same modulus, Since the modulus the dtance from the root to the origin, th acts as the radius of a circle on which i

35 cos = sin cos [cos ( 1 ) as the radius of the circle on which the roots are located The arguments of each successive root are found by repeatedly adding Thus, the roots are 1)] + i sin ( 1 )] REASONING Determine whether each statement sometimes, always, or never true Explain your reasoning 78 The p th roots of a complex number z are equally spaced around the circle centered at the origin with equally spaced around the circle 79 The complex conjugate of For any z, are real numbers Let z = a + b i and radius Find and The expression for the dtinct roots of a complex number or So, the statement always true Sample answer: If z = a + b i and or a Each root will have the same modulus, Since the modulus the dtance from the root to the origin, th acts as the radius of a circle on which the points are located The arguments of each successive root are found by repeatedly adding Thus, they are equally spaced from one and 80 OPEN ENDED Find two complex numbers a + b i in which a 0 and b 0 with an absolute value of another So, the statement always true Sample answer: The p th roots of a complex number all have the Since the absolute value of a complex number same modulus, which determine by, that acts as the radius of the circle on which the roots are located The arguments of each successive root are found by repeatedly adding Thus, the roots are equally spaced around the circle determined by =, Squaring each side results in a + b = 17 Choose a such that a as close to but less than 17 Let a = 4 Solve for b 79 The complex conjugate of For any z, are real numbers Let z = a + b i and So, 4 + i and 4 i are complex numbers with an Find and absolute value of 81 WRITING IN MATH Explain why the sum of the imaginary parts of the p dtinct p th roots of any positive real number must be zero (Hint: The roots are the vertices of a regular polygon) Sample answer: Consider the polygon created by the 8 dtinct 8th roots of 1 So,Manual the statement Cognero always esolutions - Powered by z = a + b i and and true Sample answer: If or a Page 35

36 So, 4 + i and 4 i are complex numbers with an absolute value of 81 WRITING IN MATH Explain why the sum of the imaginary parts of the p dtinct p th roots of any positive real number must be zero (Hint: The roots are the vertices of a regular polygon) For th equation, e = 4 and d = 15 4 or 375 The eccentricity and form of the equation determine that th a hyperbola with directrix x = 375 Therefore, the transverse ax of the hyperbola lies along the polar or x-ax The general equation of such a hyperbola in = rectangular form 1 The vertices lie on the transverse ax and occur when = 0 and π Sample answer: Consider the polygon created by the 8 dtinct 8th roots of 1 The vertices of the polygon in rectangular form are 1, i, and, 1,, i,, Since the roots are evenly spaced around the polygon and a vertex of the polygon lies on the positive real ax, the polygon symmetric about the real ax and the non-real complex roots occur in conjugate pairs Since the imaginary part of the sum of two complex conjugates 0, the imaginary part of the sum of all of the roots must be 0 Th will always occur when one of the roots a positive real number Write each polar equation in rectangular form 8 r = The vertices have polar coordinates (3, 0) and ( 5, π), which correspond to rectangular coordinates (3, 0) and (5, 0) The hyperbola s center the midpoint of the segment between the vertices, so (h, k) = (4, 0) The dtance a between the center and each vertex 1 The dtance c from the center to the focus at (0, 0) 4 b= or Substitute the values for h, k, a, and b into the standard form of an equation for an ellipse Write the equation in standard form For th equation, e = 4 and d = 15 4 or 375 The eccentricity and form of the equation determine that th a hyperbola with directrix x = 375 Therefore, the transverse ax of the hyperbola lies esolutions Manual Powered Cognero along the- polar or by x-ax The general equation of such a hyperbola in 83 r = Write the equation in standard form Page 36

37 84 r = The equation in standard form 83 r = Write the equation in standard form For th equation, e = 1 and d = 7 The eccentricity and form of the equation determine that th a parabola that opens horizontally with focus at the pole and a directrix x = 7 rectangular form The vertex lies between the focus F and the directrix of the parabola, occurring when correspond to rectangular coordinates = at rectangular form The vertices are the endpoints of the major ax and occur when The general equation of such a parabola in The vertex lies at polar coordinate For th equation, e = and d = 3 or The eccentricity and form of the equation determine that th an ellipse with directrix y = The general equation of such an ellipse in, which So The dtance p from the vertex to the focus at 35 Substitute the values for h, k, and p into the general equation for rectangular form The vertices have polar coordinates and, which correspond to rectangular coordinates (0, 6) and (0, ) The ellipse s center the midpoint of the segment between the vertices, so (h, k) = (0, ) The dtance a between the center and each vertex 4 The dtance c from the center to the focus at (0, 0) The equation in standard form b= 84 r = or Substitute the values for h, k, a, and b into the standard form of an equation for an ellipse Page 37

38 the center and each vertex 4 The dtance c from the center to the focus at (0, 0) b= or Substitute the values for h, k, a, and b into the standard form of an equation for an ellipse 86 x y = 1 The graph of x y = 1 a hyperbola To find the polar form of th equation, replace x with r cos and y with r sin Then simplify Identify the graph of each rectangular equation Then write the equation in polar form Support your answer by graphing the polar form of the equation 85 (x 3) + y = 9 The graph of (x 3) + y = 9 a circle with radius 3 centered at (3, 0) To find the polar form of th equation, replace x with r cos and y with r sin Then simplify Evaluate the function for several -values in its domain and use these points to graph the function The graph of th polar equation a circle Evaluate the function for several -values in its domain and use these points to graph the function The graph of th polar equation a circle 87 x + y = y The graph of x + y = y a circle To find the polar form of th equation, replace x with r cos and y with r sin Then simplify 86 x y = 1 The graph of x y = 1 a hyperbola To find the Evaluate the function for several -values in its Page 38 domain and use these points to graph the function The graph of th polar equation a circle

39 87 x + y = y 89 y x 16 = 0 Graph the equation by solving for y The graph of x + y = y a circle To find the polar form of th equation, replace x with r cos and y with r sin Then simplify Evaluate the function for several -values in its domain and use these points to graph the function The graph of th polar equation a circle 90 x + 4y + x 4y + 33 = 0 Graph the equation by solving for y Graph the conic given by each equation 88 y = x + 3x + 1 Graph the equation y = x + 3x + 1 Find the center, foci, and vertices of each ellipse 89 y x 16 = 0 Graph the equation by solving for y 91 + =1 The ellipse in standard form, where h = 8 and k Page 39 = 7 So, the center located at ( 8, 7) The ellipse has a vertical orientation, so a = 81, a = 9, and b = 9

40 The foci are c units from the center, so they are Find the center, foci, and vertices of each ellipse 91 + =1 located at ( 8, 7 ± 6 ) The vertices are a units from the center, so they are located at ( 8, 16), ( 8, ) 9 5x + 4y + 150x + 4y = 161 First, write the equation in standard form The ellipse in standard form, where h = 8 and k = 7 So, the center located at ( 8, 7) The ellipse has a vertical orientation, so a = 81, a = 9, and b = 9 Use the values of a and b to find c The equation now in standard form, where h = 3 and k = 3 So, the center located at ( 3, 3) The ellipse has a vertical orientation, so a = 5, a = 5, and b = 4 Use the values of a and b to find c The foci are c units from the center, so they are located at ( 8, 7 ± 6 ) The vertices are a units from the center, so they are located at ( 8, 16), ( 8, ) 9 5x + 4y + 150x + 4y = 161 First, write the equation in standard form The foci are c units from the center, so they are located at ( 3, 3 ± ) The vertices are a units from the center, so they are located at ( 3, ), ( 3, 8) 93 4x + 9y 56x + 108y = 484 First, write the equation in standard form The equation now in standard form, where h = 3 and k = 3 So, the center located at ( 3, 3) The ellipse has a vertical orientation, so a = 5, a = 5, and b = 4 Use the values of a and b to find c The equation now in standard form, where h = 7 and k = 6 So, the center located at (7, 6) The ellipse has a horizontal orientation, so a = 9, a = 3, and b = 4 Use the values of a and b to find c The foci are c units from the center, so they are located at ( 3, 3 ± ) The vertices are a units from the center, so they are located at ( 3, ), ( 3, esolutions Manual - Powered by Cognero 8) 93 4x + 9y 56x + 108y = 484 Page 40 The foci are c units from the center, so they are located at (7 ±, 6) The vertices are a units

41 The foci are c units from the center, so they are located at (7 ±, 6) The vertices are a units from the center, so they are located at (10, 6), (4, 6) located at ( 3, 3 ± ) The vertices are a units from the center, so they are located at ( 3, ), ( 3, 9-5 Complex Numbers and De Moivre's Theorem Solve each system of equations using Gauss 8) Jordan elimination 94 x + y + z = x + 9y 56x + 108y = 484 6x y z = 16 3x + 4y + z = 8 First, write the equation in standard form Write the augmented matrix The equation now in standard form, where h = 7 and k = 6 So, the center located at (7, 6) The Apply elementary row operations to obtain reduced row-echelon form ellipse has a horizontal orientation, so a = 9, a = 3, and b = 4 Use the values of a and b to find c The foci are c units from the center, so they are located at (7 ±, 6) The vertices are a units from the center, so they are located at (10, 6), (4, 6) Solve each system of equations using GaussJordan elimination 94 x + y + z = 1 6x y z = 16 3x + 4y + z = 8 Write the augmented matrix Apply elementary row operations to obtain reduced row-echelon form The solution (4, 0, 8) 95 9g + 7h = 30 8h + 5j = 11 3g + 10j = 73 Page 41

42 9-5 Complex The solutionnumbers (4, 0, 8) and De Moivre's Theorem Apply elementary row operations to obtain reduced row-echelon form 95 9g + 7h = 30 8h + 5j = 11 3g + 10j = 73 Write the augmented matrix Apply elementary row operations to obtain reduced row-echelon form The solution ( 1, 3, 7) 96 k n = 3p = 1 4k + p = 19 k n = 3p = 1 4k + p = 19 Page 4

43 96 k n = 9-5 3p = 1 4k + p = 19 Numbers Complex and De Moivre's Theorem k n = 3p = 1 4k + p = 19 Write the augmented matrix The solution (3, 4, 7) 97 POPULATION In the beginning of 008, the world s population was about 67 billion If the world s population grows continuously at a rate of %, the future population P, in billions, can be 00t Apply elementary row operations to obtain reduced row-echelon form predicted by P = 65e, where t the time in years since 008 a According to th model, what will be the world s population in 018? b Some experts have estimated that the world s food supply can support a population of at most 18 billion people According to th model, for how many more years will the food supply be able to support the trend in world population growth? a Let t = 10 since years since 008 Substitute t = 10 into P = 65e 00t and solve for P According to th model, the world s population will be about 794 billion b Substitute P = 18 into P = 65e t 00t and solve for According to th model, the food supply will be able to support population for about 51 more years 98 SAT/ACT The graph on the xy-plane of the The solution (3, 4, 7) 97 POPULATION In the beginning of 008, the world s about esolutions Manualpopulation - Powered bywas Cognero 67 billion If the world s population grows continuously at a rate of %, the future population P, in billions, can be 00t quadratic function g a parabola with vertex at (3, ) If g(0) = 0, then which of the following must also equal 0? A g() B g(3) C g(4) D g(6) Page 43 E g(7)

44 The correct answer F 100 FREE RESPONSE Consider the graph According tonumbers th model, the food supply will be Theorem 9-5 Complex and De Moivre's able to support population for about 51 more years 98 SAT/ACT The graph on the xy-plane of the quadratic function g a parabola with vertex at (3, ) If g(0) = 0, then which of the following must also equal 0? A g() B g(3) C g(4) D g(6) E g(7) a Give a possible equation for the function b Describe the symmetries of the graph c Give the zeroes of the function over the domain 0 π d What the minimum value of r over the domain 0 π? Since g a quadratic function, it a parabola that opens either up or down The vertex located at (3, ) Since g(0) = 0, the point (0, 0) also on the parabola Thus, the parabola opens up The parabola has an ax of symmetry at x = 3 Points to the left of the ax of symmetry will have corresponding points to the right of the ax The xcoordinate of (0, 0) units to the left of the vertex Thus, the point 3 units to the right of the vertex will have the same y-coordinate Therefore, g(3 + 3) or g(6) will have the same y-coordinate at g(0) The correct answer D 99 Which of the following expresses the complex number 0 1i in polar form? F 9(cos i sin 547) G 9(cos 55 + i sin 55) H 3(cos i sin 547) J 3(cos 55 + i sin 55) a Sample answer: The graph of a rose with 8 petals Since the graph contains the point (, 0), which lies on the x-ax, the equation will be of the form r = a cos n The petals extend units, so a = Since the number of petals even, the number of petals equal to n Thus, n = 8 or n = 4 So, a possible equation for the function r = cos 4 b Sample answer: Since the equation found in part a a function of cos, the function symmetric with respect to the polar ax It also symmetric with respect to the line = and the pole c Sample answer: Graph the rectangular function y = cos 4x over the domain 0 x π and look for where the graph intersects the x-ax Find the modulus r and argument To find a positive value for, add π So, = π or about 547 The polar form of 0 1i 9(cos i sin 547) The correct answer F 100 FREE RESPONSE Consider the graph a Give a possible equation for the function b Describe the symmetries of the graph c Give the zeroes of the function over the domain 0 π esolutions - Powered by Cognero d Manual What the minimum value of r over the domain 0 π? The graph has zeros when θ =,,,, and,,,, d Sample answer: Graph the rectangular function y = cos 4x over the domain 0 x π and look for the minimum values Page 44

45 the minimum values The minimum value of r Th occurs when =,,, and Page 45

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