In Exercises 53-64, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. 53. [4(cos 15° + i sin 15°)]³ 54. [2(cos 10° + i sin 10°)]³ 55. [2(cos 80° + i sin 80°)]³ 56. [2(cos 40° + i sin 40°)]³ + i sin 12 57. 12 + i sin 10 10 58. + i sin 6. 59. cos + i sin 18 60. cos 18 61. (1 + i) 62. (1 – i) 63. (V3 - i)* 64. (V2 - i)*

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-2 - 7i

A: z=-2-7i Now, r=-22+-72=53 & θ=tan-1yx=tan-1-7-2=π+tan-172      ∵The given complex number lies in…

Q: Define the Complex Number by using example inside this arc: (7- 3i )

A: Consider the given example of the complex number. C=7-3i The general form of the complex number is…

Q: Does the Theorem (Roots) have to be used to find the nth roots of a complex number, or can other…

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Q: For Exercises 41-42, find the indicated complex roots. Write the results in polar form using degree…

A: 41. Consider the square root of 25cos120°+isin120°. If z=rcosθ+isinθ is the given complex number in…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.√3 + i

A: 3+i

Q: Express the complex number in exact simplified rectangular form.

A:

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-4 - 4i

A: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-4 - 4i

Q: Determine the non-zero complex number, z, that satisfies the equation: 37 = z.

A: According to the given information, it is required to find non zero complex number z which satisfy…

Q: How would I find the magnitude of the complex number based on the given values for z?

A: To find the magnitude of the complex number

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.15

A: Express the complex number in trigonometric form with 0 ≤ θ < 2π.

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.1 - i

A: Given, To Evaluate, the complex number in trigonometric form with 0 ≤ θ < 2π.1 - i Here  a=1 and…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-7

A: We know that any given complex numberz=x+iy can be written in polar form as below z=rcosθ+isinθ…

Q: The polar form of complex number

A: In this question, we have to write the polar form of the complex number

Q: Use DeMoivre’s theorem to find (–√3/2 +i/2)12. Express your answer in complex form.

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Q: Find two square roots for the complex number: 25(cos120degrees +isin120degrees) answer in…

A: Given, 25cos120°+isin120°

Q: Find the product of the complex numbers z1 = cos π/4 + i sin π/4 and z2 = cos π/3 + i sin π/3. Leave…

A: To find the product of the complex numbers  z1=cosπ4+isinπ4z2=cosπ3+isinπ3

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-17

A: Let z=−17⇒z=−17+i0The complex form z=x+iy in trigonometric form is written asz=rcosθ+isinθ,…

Q: Find the cube roots of the complex number z = 125 (cos(60°) + i sin(60°)). Select all correct…

A: We have to find cube root of given complex number.

Q: Use De Moivre’s theorem to change the given complex number to the form a + bi, where a and b are…

A: Given :            -22 + 22i15   To change the given complex number to the form a + bi using De…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-5

A: Let  z = x + iy  then  z can br written as reiθ where r = x2+y2           θ = tan-1yx

Q: 4. Use DeMoivre's Theorem to find the power of the complex number [4 (cos 50 ° + i sin 50°) ]3

A: Use demovirs theorem

Q: Is it true that the quotient of two complex number is obtained by dividing their moduli and…

A: Given question says "Is it true that the quotient of two complex number is obtained by dividing…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-4√3 + 4i

A: Given: -4√3 + 4i Here  x=-43, y=4r=x2+y2=-432+42r=163+16=64r=8                        ... taking…

Q: Find the product of the complex numbers z1 = cos 210° + i sin 210° and z2 = cos 55° + i sin 55°.…

A: cosθ+i sin θ=ejθ

Q: Find the product of the complex numbers z1 = cos π/6 + i sin π/6 and z2 = cos π/4 + i sin π/4. Leave…

A: the given two complex numbers are: z1=cosπ6+isinπ6z2=cosπ4+isinπ4 we have to find the product of the…

Q: What is the complex conjugate of a complex number a  bi?

A: The given complex number is a+bi. Here a and  b  are real numbers. The real part is a and the…

Q: Find the quotient z1/z2 of the complex numbers z1 = cos 70° + i sin 70° andz2 = cos 230° + i sin…

A: The two complex numbers are given as , z1=cos70°+isin70° z2=cos230°+isin230° The value of z1z2 can…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.2 + i

A: Consider the provided information, 2+i  Express the complex number in trigonometric form with 0 ≤ θ…

Q: Show that the conjugate of the sum (product, or quotient) of two complex numbers, z1 and z2 , is the…

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Q: Let z be a complex number. Then cot(5z – n) is equal to

A: Since solve the first question for you. If you want any specific question to be solved then please…

Q: Write the complex number in rectangular form. 4( cos 210°+ i sin 210°) 4(cos 210° + i sin 210°) =…

A:

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.2√3 + 2i

A:

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.1 - 3i

A: Consider the complex number: 1-3i. Write in the trigonometric form   A complex number a+ib in the…

Q: Find the product of the complex numbers z1 = 3(cos 40° + i sin 40°) and z2 = 5(cos 70° + i sin 70°).…

A: Given complex numbers - z1 = 3cos40+isin40           ................(1)z2 = 5cos70+isin70…

Q: What is the cartesian form of the complex number: (-1-i√3)4

A: The given complex number: -1-i34.

Q: Let z = 3 – Ai, w = 1+ 2i be complex numbers. For both z and w, what is the: • Real part • Imaginary…

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Q: Find all the solutions of   18 sinz = 1.  note: z is a complex number, so please solve for the…

A: Please check step 2 for the solution.!

Q: The variable z is often used to denote a complex number and 7 is used to denote its conjugate. If z…

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Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.12

A: 12+0i

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.4 - 3i

A: Given: z=4-3i We know that any given complex numberz=a+ib can be written in polar form as below…

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-5 - 5√3i

A: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-5 - 5√3i

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.3 - 3√3i

A: Express the complex number in trigonometric form with 0 ≤ θ < 2π.3 - 3√3i = 3 (1 - √3i) Multiply…

Q: Use De Moivre’s theorem to change the given complex number to the form a + bi, where a and b are…

A: Solution: Consider the given complex number is: z = 3 + i7 We have  z = a + ib⇒z = rcos θ + i sin θ…

Q: Convert the complex number 4(cos(30°) + j sin(30°)) = Z from trigonometric form to rectangular form.…

A: The trigonometric form for a complex number is : z=r(cos θ+ jsin θ) . The rectangular form for a…

Q: What is the first step in writing the complex number -1 + i in trigonomet-ric form?

A: Consider z=x+iy is any complex number, so polar form is, z=x+iy=rcosθ+isinθWherer=x2+y2θ=π-tan-1yx…

Q: what are the complex numbers z1 and z2? for z1 i got z1= 3(cospi/12 + isinpi/2), z2= 2(cos7pi/12 +…

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Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.3 + 2i

A: Here the given complex number z=x+iy=3+2i⇒x=3,⇒y=2

Q: Express the complex number in trigonometric form with 0 ≤ θ < 2π.-2 - 2i

A: Solution: Consider the given complex number is: z = -2 - 2i The trigonometric form of a complex…