Write the complex number 4(cos 60 + i sin 60) in standard form 10. Use DeMoivre's Theorem to find (2+3i)6

Answers

Answer 1

Answer:

Answer:

a) The standard form of $z = 4\cdot (\cos 60^(\circ)+i\cdot \sin 60^(\circ))$ is $z = 2 + i\cdot 2√(3)$ , b) $z = (2+i\cdot 3)^(6) = 1219.585 + i \cdot 1829.381$ .

Step-by-step explanation:

a) The standard form of the complex number is $z = a + i\cdot b$ , $\forall \,a,b \in \mathbb{R}$ . If we get that $z = 4\cdot (\cos 60^(\circ)+i\cdot \sin 60^(\circ))$ , whose standard form is obtained by algebraic means:

1) $z = 4\cdot (\cos 60^(\circ)+i\cdot \sin 60^(\circ))$ Given

2) $z = (4\cdot \cos 60^(\circ))+i\cdot (4\cdot \sin 60^(\circ))$ Distributive and Associative properties.

3) $z = 2 + i\cdot 2√(3)$ Multiplication/Result.

The standard form of $z = 4\cdot (\cos 60^(\circ)+i\cdot \sin 60^(\circ))$ is $z = 2 + i\cdot 2√(3)$ .

b) The De Moivre's Theorem states that:

$z = (a+i\cdot b)^(n)= r^(n)\cdot (\cos \theta + i\cdot \sin \theta)$

Where:

$r =\sqrt{a^(2)+b^(2)}$ and $\theta = \tan^(-1) \left((b)/(a)\right)$ .

If we know that $z = (2+i\cdot 3)^(6)$ , then:

$r = \sqrt{2^(2)+3^(2)}$

$r =√(13)$

$r \approx 3.606$

$\theta = \tan^(-1)\left((3)/(2) \right)$

$\theta \approx 56.310^(\circ)$

The resulting expression is:

$z = 3.606^(6)\cdot (\cos 56.310^(\circ)+i\cdot \sin 56.310^(\circ))$

$z = 1219.585+i\cdot 1829.381$

Therefore, $z = (2+i\cdot 3)^(6) = 1219.585 + i \cdot 1829.381$ .

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23 greater than b is at least -276

Answers

Answer:

23 + b ≤ -276

Step-by-step explanation:

In this exercise, you're required to write an algebraic expression for the word problem. Thus, you'll write out a mathematical equation using the given values and unknown variable.

Translating the word problem into an algebraic expression, we have;

23 + b ≤ -276

Simplifying further, we have;

b ≤ -276 - 23

b ≤ -299

When hired at a new job selling jewelry, you are given two pay options:Option A: Base salary of $16,000 a year, with a commission of 7% of your sales

Option B: Base salary of $25,000 a year, with a commission of 2% of your sales

In order for option A to produce a larger income, you would need sell at least $_______ of jewelry each year

Answers

You are required to sell a minimum of 180,001 of jewelry each year.

Given that,

The base salary is $16,000 per year and there is a 7% commission on sales for option A.
The base salary is $25,000 per year and there is a 2% commission on sales for option B.
Also, let us assume the sales be x

Based on the above information, the following expression should be made:

16,000 + 0.07x > 25,000 + 0.02x

0.07x - 0.02x > 25,000 - 16,000

0.05x > 9000

x > 9000 ÷ 0.05

x > 180,000

Therefore we can conclude that you are required to sell a minimum of 180,001 of jewelry each year.

Learn more: brainly.com/question/12127730

16,000 + 0.07x > 25,000 + 0.02x
0.07x - 0.02x > 25,000 - 16,000
0.05x > 9000
x > 9000 / 0.05
x > 180,000....so they would need to sell at least 180,001

Plz help I will mark brainlest!

Answers

number 12 : answer is b

Answer:

the first picture is 5.2 so b

The second picture is 2 and a half centimeters so also b

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Step-by-step explanation:

Determine if the sequence below is arithmetic or geometric and determine thecommon difference / ratio in simplest form.
100, 20, 4, ...

Answers

Answer:

its a geometric progression. r=1/5 or 0.2

Reese read twice as many pages Saturday than she read Sunday. If she read a total of 78 pages over the weekend, how many pages did Reese read Sunday?

Answers

Answer:

Reese read 26 pages on Sunday.

Step-by-step explanation:

If she read twice the amount on Saturday than she read on Sunday, then you can replace the days with variables. 2x for Saturday and x for Sunday. That leaves you with 2x + x = 78. This can be simplified to 3x = 78. We can solve this by dividing both sides by 3. This leaves you with x = 26. If x represents the pages read on Sunday then Reese read 26 pages on Sunday.

Find the missing variable in the following question.If there is direct variation and y = 35 and x = 14, find x when y = 15.

A. x = 15
B. x = 6
C. x = 24
D. x = 12

Answers

Answer:

X=2×3=6

Step-by-step explanation:

Y=5, x=2×3=6

I hopethis helps

BRAINLIEST PLEASE

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Will mark brainest help please !!!