Answer:
See below for answers and explanations
Step-by-step explanation:
Problem 1
Let [tex]z=r(\cos\theta+i\sin\theta)[/tex] be a complex number, where [tex]n[/tex] is an integer and [tex]n\geq 1[/tex]. By DeMoivre's Theorem, if [tex]z^n=r^n(\cos\theta+i\sin\theta)^n[/tex], then [tex]z^n=r^n(\cos n\theta+i\sin n\theta)[/tex].
Hence, we must first convert from rectangular to polar form:
[tex]z=-1-i\\\\r=\sqrt{a^2+b^2}\\\\r=\sqrt{(-1)^2+(-1)^2}\\\\r=\sqrt{1+1}\\\\r=\sqrt{2}[/tex]
[tex]\displaystyle \theta=\tan^{-1}\biggr(\frac{y}{x}\biggr)\\\\\theta=\tan^{-1}\biggr(\frac{-1}{-1}\biggr)\\ \\\theta=\tan^{-1}(1)\\\\\theta=45^\circ[/tex]
Therefore, the rectangular form for the complex number is [tex]z=\sqrt{2}(\cos45^\circ+i\sin45^\circ)[/tex], and now we can apply DeMoivre's Theorem to raise the complex number to the 10th power:
[tex]z=\sqrt{2}(\cos45^\circ+i\sin45^\circ)\\\\z^{10}=\sqrt{2}^{10}(\cos45^\circ+i\sin45^\circ)^{10}\\\\z^{10}=2^5(\cos10\cdot45^\circ+i\sin10\cdot45^\circ)\\\\z^{10}=32(\cos450^\circ+i\sin450^\circ)\\\\z^{10}=32(0+i)\\\\z^{10}=32i[/tex]
Therefore, [tex]z^{10}=32i[/tex] in rectangular form.
Problem 2
[tex]3\:\text{cis}\:90^\circ=3(\cos90^\circ+i\sin90^\circ)=3(0+i)=3i[/tex]
Therefore, [tex]3\:\text{cis}\:90^\circ=3i[/tex] in rectangular form.
50 POINTS PLEASE HELP What is the volume of this prism???
Answer:
128
Step-by-step explanation:
multiply length, width, and hight.
[Not including the "triangle roof"] 128 in²
V = L * W * H
V = 8 in * 4 in * 4 in
V = 128 in²
[Including the "triangle roof"] 176in²
V = [tex]\frac{ L * W * H}{2}[/tex]
V = [tex]\frac{8\text{ in }*\,4\text{ in }*\,3\text{ in }}{2}[/tex]
V = [tex]\frac{96\;in^2}{2}[/tex]
V = 48 in²
-
48 in² + 128 in² = 176in²
Use the substitution method to determine the solution to the system of equations below. y-2x+2=1 6x - 3y = 3
Answer:
y = 2x - 1
Step-by-step explanation:
y - 2x + 2 = 1
6x - 3y = 3
[tex]y - 2x + 2 = 1 \\ +2x - 2 +2x - 2\\y = 2x - 1[/tex]
Plug in for y:
6x - 3(2x - 1) = 3
6x - 6x + 3 = 3
Simplified, 3 = 3
Answer: this solution has infinite solutions
Step-by-step explanation:
original system:
y-2x+2 = 1
6x-3y = 3
Move 2 to right side of equation
y -2x = -1
-3y+6x = 3
solve for y in first equation
⇒ y = 2x - 1
insert y into second equation
⇒ -3(2x-1) + 6x = 3
simplify
⇒ -6x+3+6x = 3
simplify once more
⇒ 3 = 3
this solution has infinite solutions
Find the area of the figure.
Answer:
Area = 38 ft^2
Step-by-step explanation:
Divide two parts of a rectangle.
Big Rect: A = 10 ft * 3 ft = 30 ft^2
Small Rect: A = 2 ft * 4 ft =8 ft^2
Total Area = 30 + 8 = 38 ft^2
Find the slope, if it exists, of the line containing the pair of points (1, -2) and (1, - 4).
Answer:
no answer
Step-by-step explanation:
1, -2 ; 1, -4 : -4-2/1-1 = no solution
what is :
2 tons + 10 pounds
And
4 tons + 10 pounds
and then explain it properly
2 tons + 10 pounds = 4,010 pounds
4 tons + 10 pounds = 8,010 pounds
Step-by-step explanation:
2,000 POUNDS EQUALS 1 TON.
a.) 2 tons = 2,000 X 2 = 4,000 pounds
4,000 pounds + 10 pounds = 4,010 pounds
2 tons + 10 pounds = 4,010 pounds
b.) 4 tons = 2,000 X 4 = 8,000 pounds
8,000 pounds + 10 pounds = 8,010 pounds
4 tons + 10 pounds = 8,010
This is had can you help me
Answer:
#9 is 6 #10 is 1 #11 is 7 #12 is 6
Step-by-step explanation:
There are 24 pictures on a roll of film. Adelle had 84 pictures from her trip home. She said that 12 other pictures were spoiled. How many rolls of film did Adelle use?
Answer:
Total pictures = 84+12 = 96
roll of films = 96 / 24 = 4
she used 4 rolls
4. (1) Solve forx: 13 11 3x+16 65
Step-by-step explanation:
13113x+1665
_____________________________
Rewrite 13113 as 9 . 1457
Rewrite 1665 as 9 . 185
= 9 . 1457x + 9 . 185
=9(1457x + 185)
what transformation of figure 1 results and figure 2?
reflection
dialation
translation
what are the zeros of this function? f(x) = 3x^2 + 2x - 8
Answer:
4/3, -2
Step-by-step explanation:
find x ...............
In the first circle, x is the radius and also the hypotenuse of a right triangle with legs measuring 9 and 12. So by the Pythagorean theorem,
x² = 9² + 12² = 225 = 15² ⇒ x = 15
In the second circle, the chords AB and CD have the same length, and each of AP, BP, CP, and DP are radii of the circle and also have the same length. This means ∆ABP and ∆CDP are congruent, so the two labeled angles are also congruent, so x = 54.
That’s the Answer in the right order
Answer:
3rd one
Step-by-step explanation:
i hope have a great day
.
A jar contains 11 green marbles, 6 red marbles, and 9 blue marbles. A marble is selected at random, not replaced, and then a second marble is selected. What is the probability of selecting a red marble followed by a green marble?
Answer:
red marbel
[tex] = \frac{3}{13} [/tex]
green marbel
[tex] = \frac{9}{26} [/tex]
Step-by-step explanation:
for the red marbel you first add all the number of balls given and the total is 26 then you take the number of red marbels dividing the total number of all the balls and if you are able to cancel out you can and there you get your answer same applies to the green marbel
In a certain family, the probability of having twins in 0.08 and the probability of having a single baby is 0.92.
What is the probability of a family having two sets of twins?
Answer: SE = √. pA(1 - pA). nA. +. pB(1 - pB). nB. = √. 0.08(0.92). 200. +. 0.344(0.656). 200. = 0.039. (b) The sample sizes of 200 are large enough
Step-by-step explanation:
HELP for (ii)…..how do we find a?
By using the information given on the graph, we can see that a = 3.
How to find the value of a?
Here we have an exponential of the form:
[tex]y = p*a^x[/tex]
And by looking at the graph, it passes through (0, 4) and (2, 36), so we can write:
[tex]4 = p*a^0\\\\36 = p*a^2[/tex]
From the first equation we get that p = 4, replacing that on the second one we get:
[tex]36 = 4*a^2\\\\36/4 = 9 = a^2\\\\\sqrt{9} = 3 = a[/tex]
So we conclude that a = 3.
If you want to learn more about exponential equations:
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One number is ten more than twice another. Their sum is one. Find the numbers.
answer:
x = 4
y = -3
steps :
x = 10 + 2y
x + y = 1
2y = x - 10 -> y = 1/2x - 5
x + y = 1 -> y = 1 - x
since theyre both equal to y they are equal to each other
1/2x - 5 = 1 - x
3/2x = 6
x = 4
plug 4 back where x is
y = 1 - 4
y = -3
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Evaluate the determinant
What is true about open-end credit? OA It is not typically used for a single purchase but for many purchases over a period of time. B. This type of credit is sometimes referred to as a consumer loan. OC. This type of credit is commonly used for big ticket things like mortgages or car loans. Payments typically stay the same amount from month to month.
Answer:
Option B
Step-by-step explanation:
Open end credit 's biggest example is our credit card
You use credit card and take the money borrow from bank as per your requirement.If you pay in time your credit score increases else decreases.Option B is correct
help will mark Brainliest!!
Answer:
1/20
Step-by-step explanation:
1/4 = 0.25
0.25 / 5 = 0.05
0.05 as a fraction = 5/100
simplified
5 / 5 = 1
100 / 5 = 20
1/20
In the given right triangle, write all trigonometric ratios ?
Answer:
see below
Step-by-step explanation:
You will need the hypotenuse 12 ^2 + 9^2 = h^2 h = 15
In a right triangle:
S O H C A H T O A
sin s = S A H = Opposite / Hypotenuse = 12/15
cos r = C A H = Adjacent / Hypotenuse = 12/ 15
Tan s = T O A = 12 / 9
cos (theta) = cos s = 9 / 15
Tan r = 9/12
arc cos(theta) = theta = arccos(9/15) = 53.13 degrees
Ms. Wall will roll a dice. What is
the probability that she will roll an even number?
Answer:
50%
Step-by-step explanation:
half of the numbers on dice are even
PLEASE HELP 25 POINTS!!!
Sequence: 2 + 1 + 1/2 + 1/4 + 1/8 + ...
Explanation:
Here Given Sum to Infinity
[tex]\sf S \infty \ =a_1 \ x \ \dfrac{1}{1-r}[/tex]
Identify following
a1 = 2r = 1/2Solve:
[tex]\rightarrow \sf S \infty \ =2 \ x \ \dfrac{1}{1-\frac{1}{2} }[/tex]
[tex]\rightarrow \sf S \infty \ =4[/tex]
The sum of this infinity series is 4
Sophia is 12 years old. Her Uncle Reynald tells her that if she adds 5 to her age, multiplies the sum by 3, and then subtracts 4 from the product, she will find his age. She tells him that his age equals the expression 12 + 5 × 3 − 4 . Is Sophia correct? Choose the correct answers from the drop-down menus. Sophia is Choose... . Her uncle's actual age is Choose... .
Answer:
Yes, Sophia is correct
Step-by-step explanation:
The expression is correct and if you solve it, you get 23 which should be her uncle's age.
Can a triangle have sides with the given lengths?
2 in, 4 in, 6 in
Answer:
NO
Step-by-step explanation:
Because the sum of the minor sides equals the measure of the major side, it must be less.
2 + 4 = 6
Hope this helps
The measure of an angle is 60 greater than the measure of its complement find the measure of each angle
Answer:
[tex] \\ \\ 90-60=30degree \\ \\ [/tex]
Answer:
15° and 75°
Step-by-step explanation:
The sum of the measures of complementary angles is 90.
Angle 1 measures x.
Angle 2 measures x + 60.
Their sum is 90.
x + x + 60 = 90
2x + 60 = 90
2x = 30
x = 15
x + 60 = 75
Answer: 15° and 75°
Select the statement that describes this expression: fraction 1 over 2 x (734 − 246).
Half the sum of 734 and 246
fraction 1 over 2 the difference between 734 and 246
fraction 1 over 2 the quotient of 734 and 246
2 times the difference between 734 and 246
The perimeter of a rectangle is 84cm if the width of rectanle is 16 cm Find its lenth.
Look at the diagram. Which angle is supplementary to A. B. C. D.
The angles supplementary. to ∠DOC are ∠DOA and ∠COB.
What is supplementary angle?Two angles are called supplementary when their measures add up to 180 degrees.
1. Take, ∠COB:
So, ∠DOC+ ∠COB = ∠DOB = 180°. (Linear pair)
Hence, ∠DOC and ∠COB are supplementary.
2. Now, take angle DOA:
∠DOC + ∠DOA = ∠COA = 180°. (Linear pair)
Thus, ∠DOC and ∠DOA are supplementary.
Hence, the angles supplementary. to ∠DOC are ∠DOA and ∠COB.
Learn more about supplementary angle here:
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Given that circumference of a circle is 44m. Calculate it's area
Answer:
Area=154.062 [tex]m^{2}[/tex]
Step-by-step explanation:
As Circumference of a circle is given by;
[tex]C=2\times \pi\times r[/tex]
As the circumference is 44m so we can calculate the value of r by using the above formula.
[tex]44=2\times \pi \times r\\\\\frac{44}{2\pi} =r\\\\\implies r=\frac{22}{\pi} =7.003 m[/tex]
Now to Calculate the Area of a Circle.
As the area of a circle is,
[tex]Area =\pi \times r^{2}\\ Area=\pi \times (7.003)^{2} \\Area=\pi \times 49.039\\Area=154.062m^{2}[/tex]
Help urgent need this help please
Answer:
First option is the correct answer
[tex] A(\triangle ABC) =572\: yd^2[/tex]
Step-by-step explanation:
Given scale is: 3 cm = 6 yd [tex] \implies[/tex] 1 cm = 2 ydBased upon the above scale, we find:Base of the triangle (AC) = 13 cm = 26 ydHeight of the triangle (h) = 22 cm = 44 yd[tex] A(\triangle ABC) = \frac{1}{2} * AC*h[/tex][tex] \implies A(\triangle ABC) = \frac{1}{2} (26)(44)[/tex][tex] \implies A(\triangle ABC) =572\: yd^2[/tex]