1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Natali5045456 [20]
3 years ago
6

Simplify the complex numbers using de Moivre's Theorem and match them with their solutions.

Mathematics
1 answer:
alexandr1967 [171]3 years ago
4 0

For the numbers in a+bi form, convert to polar form:


1+i=\sqrt2\dfrac{1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)

By DeMoivre's theorem,

(1+i)^5=(\sqrt2)^5\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=4\sqrt2\dfrac{-1-i}{\sqrt2}=-4-4i


-1+i=\sqrt2\dfrac{-1+i}{\sqrt2}=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4}\right)

\implies(-1+i)^6=(\sqrt2)^6\left(\cos\dfrac{18\pi}4+i\sin\dfrac{18\pi}4\right)=8i


\sqrt3+i=2\dfrac{\sqrt3+i}2=2\left(\cos\dfrac\pi6+i\sin\dfrac\pi6\right)

\implies2(\sqrt3+i)^{10}=2^{11}\left(\cos\dfrac{10\pi}6+i\sin\dfrac{10\pi}6\right)=2^{11}\dfrac{1-i\sqrt3}2=2^{10}(1-i\sqrt3)


For the numbers already in polar form, DeMoivre's theorem can be applied directly:


2\left(\cos20^\circ+i\sin20^\circ\right)^3=2\left(\cos60^\circ+i\sin60^\circ\right)=2\dfrac{1+i\sqrt3}2=1+i\sqrt3


2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)^4=2(\cos\pi+i\sin\pi)=-2


At second glance, I think the 2s in the last two numbers should also be getting raised to the 3rd and 4th powers:


\left(2(\cos20^\circ+i\sin20^\circ)\right)^3=8\left(\cos60^\circ+i\sin60^\circ\right)=4+4\sqrt3

\left(2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)\right)^4=16(\cos\pi+i\sin\pi)=-16

You might be interested in
Factor the expression by taking out the greatest common factor.<br><br> 9x^2y - 6xy^2
Tcecarenko [31]

The greatest common factor of the two terms is  3 x y .

When factored, the expression is

                                 3 x y (3x - 2y)
3 0
3 years ago
Let f(x)=2x^2-x-1 and let g(x)=x-1 Which statement is true?​
sergij07 [2.7K]

Answer:

B is true

Step-by-step explanation:

given f(x) = 2x² - x - 1 and g(x) = x - 1

A

If x = - 1 is a root then f(- 1) = 0

f(- 1) = 2(- 1)² - (- 1) - 1 = 2 + 1 - 1 = 2 ≠ 0 ← False

B

If (x - 1) is a factor then x = 1 is a root and f(1) = 0

f(1) = 2(1)² - 1 - 1 = 2 - 1 - 1 = 0

⇒ g(x) = x - 1 is a factor of f(x) ← True

C

If x = 2 is a root then f(2) = 0

f(2) = 2(2)² - 2 - 1 = 8 - 3 = 5 ≠ 0 ← False

D

\frac{x2x^2-x-1}{x-1} = \frac{(2x+1)(x-1)}{x-1}

Cancel the factor (x - 1) on the numerator/ denominator, leaving

\frac{f(x)}{g(x)} = 2x + 1 with remainder 0 ≠ 2 ← False

7 0
3 years ago
Please answer the area please
Nata [24]

The area of a trapezoid is:

A = \frac{a + b}{2} h

A = area

a = base 1

b = base 2

h = height

Area = unknown

base 1 = 2.5

base 2 = 7.5

height = 2.5

^^^Plug these numbers into the formula!

A = (\frac{2.5 + 7.5}{2} )(2.5)

A = (10/2)(2.5)

A = (5)(2.5)

A = 12.5

Hope this helped!

4 0
3 years ago
The drawing plan for an art studio shows a rectangle that is 17.2 inches by 6 inches. The scale in the plan is 2 in.: 5 ft. Find
Norma-Jean [14]

Answer:

The length is 43 feet, the width is 15 feet, and the area is 645 square feet

Step-by-step explanation:

Let us solve the question

∵ The drawing plan shows a rectangle that is 17.2 inches by 6 inches

∴ The drawing length = 17.2 inches

∴ The drawing width = 6 inches

∵ The scale in the plan is 2 in.: 5 ft

→ By using the ratio method

→  inches: feet

→  2         : 5

→  17.2     : L

→  6         : W

→ By using the cross multiplication

∵ 2 × L = 17.2 × 5

∴ 2L = 86

→ Divide both sides by 2

∴ L = 43 feet

∴ The actual length is 43 feet

→ By using the cross multiplication

∵ 2 × W = 6 × 5

∴ 2W = 30

→ Divide both sides by 2

∴ W = 15 feet

∴ The actual width is 15 feet

∵ The area of the rectangle is A = L × W

∵ L = 43 feet

∵ W = 15 feet

→ Substitute them in the rule of the area

∴ A = 43 × 15

∴ A = 645 feet²

∴ The actual area is 645 square feet

7 0
3 years ago
You gain 60 points in a board game negative or positive integer
Mkey [24]
It is a positive integer
5 0
3 years ago
Read 2 more answers
Other questions:
  • You make $1 per quart picking strawberries. Do the ordered pairs (number of quarts, earnings) represent a function? Explain your
    9·1 answer
  • The product of n and 5
    5·1 answer
  • What are the period and amplitude of the function?
    11·1 answer
  • 5.6÷2×2×2+(12.75 + 7.45)​
    9·1 answer
  • Louise has a television in her family room that is 3 feet wide and 5 feet long. She has a television that is similar in shape in
    5·1 answer
  • What is remainder one as a fraction?
    11·2 answers
  • Ben has 43 coins belonging to dimes or quarters. If he has
    6·1 answer
  • 1) If 2/7gallon of paint covers 1/5 of a wall, then how much paint is needed to cover
    10·1 answer
  • A) x²-4x+3=0 prove in factorization form<br>​
    13·1 answer
  • An amusement park offers ride tickets for $1. 50 each to customers who buy the quantities of tickets indicated in the graph belo
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!