100 POINTS AND BRAINLIEST FOR THIS WHOLE SEGMENT
a) Find zw, Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
b) Find z^10. Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
c) Find z/w. Write your answer in both polar form with ∈ [0, 2pi] and in complex form.
d) Find the three cube roots of z in complex form. Give answers correct to 4 decimal
places.
Answers
Answer:
See Below (Boxed Solutions).
Step-by-step explanation:
We are given the two complex numbers:
[tex]\displaystyle z = \sqrt{3} - i\text{ and } w = 6\left(\cos \frac{5\pi}{12} + i\sin \frac{5\pi}{12}\right)[/tex]
First, convert z to polar form. Recall that polar form of a complex number is:
[tex]z=r\left(\cos \theta + i\sin\theta\right)[/tex]
We will first find its modulus r, which is given by:
[tex]\displaystyle r = |z| = \sqrt{a^2+b^2}[/tex]
In this case, a = √3 and b = -1. Thus, the modulus is:
[tex]r = \sqrt{(\sqrt{3})^2 + (-1)^2} = 2[/tex]
Next, find the argument θ in [0, 2π). Recall that:
[tex]\displaystyle \tan \theta = \frac{b}{a}[/tex]
Therefore:
[tex]\displaystyle \theta = \arctan\frac{(-1)}{\sqrt{3}}[/tex]
Evaluate:
[tex]\displaystyle \theta = -\frac{\pi}{6}[/tex]
Since z must be in QIV, using reference angles, the argument will be:
[tex]\displaystyle \theta = \frac{11\pi}{6}[/tex]
Therefore, z in polar form is:
[tex]\displaystyle z=2\left(\cos \frac{11\pi}{6} + i \sin \frac{11\pi}{6}\right)[/tex]
Part A)
Recall that when multiplying two complex numbers z and w:
[tex]zw=r_1\cdot r_2 \left(\cos (\theta _1 + \theta _2) + i\sin(\theta_1 + \theta_2)\right)[/tex]
Therefore:
[tex]\displaystyle zw = (2)(6)\left(\cos\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right) + i\sin\left(\frac{11\pi}{6} + \frac{5\pi}{12}\right)\right)[/tex]
Simplify. Hence, our polar form is:
[tex]\displaystyle\boxed{zw = 12\left(\cos\frac{9\pi}{4} + i\sin \frac{9\pi}{4}\right)}[/tex]
To find the complex form, evaluate:
[tex]\displaystyle zw = 12\cos \frac{9\pi}{4} + i\left(12\sin \frac{9\pi}{4}\right) =\boxed{ 6\sqrt{2} + 6i\sqrt{2}}[/tex]
Part B)
Recall that when raising a complex number to an exponent n:
[tex]\displaystyle z^n = r^n\left(\cos (n\cdot \theta) + i\sin (n\cdot \theta)\right)[/tex]
Therefore:
[tex]\displaystyle z^{10} = r^{10} \left(\cos (10\theta) + i\sin (10\theta)\right)[/tex]
Substitute:
[tex]\displaystyle z^{10} = (2)^{10} \left(\cos \left(10\left(\frac{11\pi}{6}\right)\right) + i\sin \left(10\left(\frac{11\pi}{6}\right)\right)\right)[/tex]
Simplify:
[tex]\displaystyle z^{10} = 1024\left(\cos\frac{55\pi}{3}+i\sin \frac{55\pi}{3}\right)[/tex]Simplify using coterminal angles. Thus, the polar form is:
[tex]\displaystyle \boxed{z^{10} = 1024\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)}[/tex]
And the complex form is:
[tex]\displaystyle z^{10} = 1024\cos \frac{\pi}{3} + i\left(1024\sin \frac{\pi}{3}\right) = \boxed{512+512i\sqrt{3}}[/tex]
Part C)
Recall that:
[tex]\displaystyle \frac{z}{w} = \frac{r_1}{r_2} \left(\cos (\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)\right)[/tex]
Therefore:
[tex]\displaystyle \frac{z}{w} = \frac{(2)}{(6)}\left(\cos \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right) + i \sin \left(\frac{11\pi}{6} - \frac{5\pi}{12}\right)\right)[/tex]
Simplify. Hence, our polar form is:
[tex]\displaystyle\boxed{ \frac{z}{w} = \frac{1}{3} \left(\cos \frac{17\pi}{12} + i \sin \frac{17\pi}{12}\right)}[/tex]
And the complex form is:
[tex]\displaystyle \begin{aligned} \frac{z}{w} &= \frac{1}{3} \cos\frac{5\pi}{12} + i \left(\frac{1}{3} \sin \frac{5\pi}{12}\right)\right)\\ \\ &=\frac{1}{3}\left(\frac{\sqrt{2}-\sqrt{6}}{4}\right) + i\left(\frac{1}{3}\left(- \frac{\sqrt{6} + \sqrt{2}}{4}\right)\right) \\ \\ &= \boxed{\frac{\sqrt{2} - \sqrt{6}}{12} -\frac{\sqrt{6}+\sqrt{2}}{12}i}\end{aligned}[/tex]
Part D)
Let a be a cube root of z. Then by definition:
[tex]\displaystyle a^3 = z = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]
From the property in Part B, we know that:
[tex]\displaystyle a^3 = r^3\left(\cos (3\theta) + i\sin(3\theta)\right)[/tex]
Therefore:
[tex]\displaystyle r^3\left(\cos (3\theta) + i\sin (3\theta)\right) = 2\left(\cos \frac{11\pi}{6} + i\sin \frac{11\pi}{6}\right)[/tex]
If two complex numbers are equal, their modulus and arguments must be equivalent. Thus:
[tex]\displaystyle r^3 = 2\text{ and } 3\theta = \frac{11\pi}{6}[/tex]
The first equation can be easily solved:
[tex]r=\sqrt[3]{2}[/tex]
For the second equation, 3θ must equal 11π/6 and any other rotation. In other words:
[tex]\displaystyle 3\theta = \frac{11\pi}{6} + 2\pi n\text{ where } n\in \mathbb{Z}[/tex]
Solve for the argument:
[tex]\displaystyle \theta = \frac{11\pi}{18} + \frac{2n\pi}{3} \text{ where } n \in \mathbb{Z}[/tex]
There are three distinct solutions within [0, 2π):
[tex]\displaystyle \theta = \frac{11\pi}{18} , \frac{23\pi}{18}\text{ and } \frac{35\pi}{18}[/tex]
Hence, the three roots are:
[tex]\displaystyle a_1 = \sqrt[3]{2} \left(\cos\frac{11\pi}{18}+ \sin \frac{11\pi}{18}\right) \\ \\ \\ a_2 = \sqrt[3]{2} \left(\cos \frac{23\pi}{18} + i\sin\frac{23\pi}{18}\right) \\ \\ \\ a_3 = \sqrt[3]{2} \left(\cos \frac{35\pi}{18} + i\sin \frac{35\pi}{18}\right)[/tex]
Or, approximately:
[tex]\displaystyle\boxed{ a _ 1\approx -0.4309 + 1.1839i,} \\ \\ \boxed{a_2 \approx -0.8099-0.9652i,} \\ \\ \boxed{a_3\approx 1.2408-0.2188i}[/tex]
Related Questions
Jerome solved the equation below by graphing. log Subscript 2 Baseline x + log Subscript 2 Baseline (x minus 2) = 3 Which of the following shows the correct system of equations and solution?
Answers
Answer:
B. on edge 2021
Step-by-step explanation:
y 1 = StartFraction log x Over log 2 EndFraction + StartFraction log (x minus 2) Over log 2 EndFraction, y 2 = 4; x = 4
Answer:
It's B :))
Step-by-step explanation:
WHERE ARE THE EXPERTS AND ACE!!!!!!! I NEED HELP PLS SHARE YO SMARTNESS!!!!! WILL GIVE BRAINLIEST AND RATE AND VOTE!!!
Answers
Answer:
C and A
Step-by-step explanation:
C and A are correct answers
Please help geometry
Answers
Answer:
1187.52
Step-by-step explanation:
2 π r h + 2 π r² = 2·π·7·20+2·π·7² ≈ 1187.52202
Here you go i hope this will help you :)Provide a real-world situation in which it would be appropriate to use a geometric sequence or series. Be sure to explain why a series or a sequence is more appropriate
Answers
Step-by-step explanation:
Well talking about the practical use of Arthematic sequence or series..
i have an example..
suppose there is a highway and in most condition.. Petrol or diesel station are kept in a common distance..
now you can easily calculate or predict where the next petrol or diesel station is ..
and about Geometric sequence or series
There are many uses of geometric sequences in everyday life, but one of the most common is in calculating interest earned. Mathematicians calculate a term in the series by multiplying the initial value in the sequence by the rate raised to the power of one less than the term number. The sequence allows a borrower to know the amount his bank expects him to pay back using simple interest
and
Physicists use geometric progressions to calculate the amount of radioactive material left after any given number of half-lives of the material. During each half-life, the material decays by 50 percent.
Find the length of x. Assume the triangles are similar.
A. 25
B. 12
C. 20
D. 19
Answers
Answer:
C=20
Step-by-step explanation:
The value of x is 20. so option C is correct.
What are corresponding sides?In literal sense, corresponding means pair wise sides. Like the right corner sides of two triangles etc.
For two similar figures, the pair by pair similar sides of those two similar figures are called corresponding sides. They are of same measurement.
Find the length of x. Let Assume the triangles are similar.
x/ 12 = 10/ 6
Cross multiplication;
x/ 12 = 10/ 6
x = 12 ( 10/ 6)
x = 20
Hence, the value of x is 20. so option C is correct.
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Helpppppppp???????????????
Answers
Answer:
A, [tex]{x | x<-2}[/tex]
Step-by-step explanation:
We have the inequality as
[tex]3x<-6[/tex]
We can simplify this inequality a little bit more by dividing 3 from both sides.
That gets us
[tex]x<-2[/tex]
We can see the only answer that has this is A.
Which possible buffet prices could Noah could charge and still maintain the restaurant owner’s revenue requirements?
Select the correct prices in the table.
$12 $13 $14 $15
Answers
Answer:
The answer is actually 12 and 13.
Step-by-step explanation:
You have to plug the number in to see what is greater than or equal to 130. 12 is equal to 160, so 12 works. 13.00 is equal to 130. 14 is equal to 112 and 15 is 90.
The possible buffet prices could noah could charge and still maintain the restaurant owner’s revenue requirements is actually 12 and 13.
We have given that the table,
What is the prices?Pricing is the process whereby a business sets the price at which it will sell its products and services, and may be part of the business's marketing plan.
You have to plug the number in to see what is greater than or equal to 130. 12 is equal to 160, so 12 works. 13.00 is equal to 130. 14 is equal to 112 and 15 is 90.
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Please help out need explanation so I can know how to do it
Answers
Answer:
B = 60in^2
H = 4in
P = 34in
SA = 280
Step-by-step explanation:
Base - The base I assume would be the red rectangle, and 12x5 would be the area of that, so the answer is 60
Height - I assume that the height is 12, if the prism is vertical, if horizontal, then the height would be 4.
Perimeter - The length of the side lengths, so the base is the red sqaure 12+12+5+5=34
Surface Area - Base=60*4 (There are 4 "bases") =240, and then we have 5*4 (For the little squares, and times this by 2 becasue there are 2 of them) = 20*2=40
40+240=280
Explain why having three congruent angles in two triangles is not a good way to prove two
triangles are congruent. Come up with a counter example, you can describe or draw it.
Answers
Draw an equilateral triangle with side lengths of 1. Each angle here is 60 degrees, which is true of any equilateral triangle.
Now draw another equilateral triangle that has side lengths of 2 units. Clearly this triangle is not the same size as the previous one, but the angles are all still 60 degrees.
We have an example in which there are 2 triangles with the same angles, but the triangles are not congruent. Therefore, having info about congruent angles only isn't sufficient to prove triangles to be congruent.
Sum of zeroes of polynomial 2x²-x-3 is ? Full solved answers ✔ No spam ❌
Answers
Answer:
[tex]\frac{1}{2}[/tex]
Step-by-step explanation:
Given a quadratic equation in standard form
ax² + bx + c = 0 , then the sum of the roots is
α + β = - [tex]\frac{b}{a}[/tex]
Here 2x² - x - 3 = 0 is in standard form
with a = 2 and b = - 1 , then
sum of roots = - [tex]\frac{-1}{2}[/tex] = [tex]\frac{1}{2}[/tex]
help pls and thank you :)
Answers
Answer: x+2y+6=0
Option 3
Explanation:
y = (-x-6)/2
2y = -x-6
x+2y+6=0
Must click thanks and mark brainliest
please I need help here
Answers
Answer:
90 degrees
Step-by-step explanation:
The sum of interior angles of every triangle is 180 degrees
Luckily 90 degrees has already been represented on the triangle
SOLUTION
x+90°=180°
x=180°-90°
x= 90°
Therefore, the answer is 90°
Find an equation for y (in terms of β and α). Then find tan y.
Answers
Answer:
See belowStep-by-step explanation:
Angle opposite to 6 is complementary of α:
90 - αSimilarly, angle opposite to side 4 of the triangle in the right side is:
90 - βSum of the interior angles of the triangle on the bottom:
90 - α + 90 - β + γ = 180γ = α + βtan γ = tan (α + β)From the smaller triangles we get:
tan α = 4/6 = 2/3tan β = 3/4Use the identity below and find the value of tan γ:
tan (α + β) = (tan α + tan β)/(1 - tan α tan β)tan γ = tan (α + β) = (2/3 + 3/4) / (1 - 2/3*3/4) =(17/12) / (1/2) = 17/6 = 2 5/6What part of an hour passes between 2:24pm and 4:40pm
Answers
Answer:
2 1/4 hrs
Step-by-step explanation:
2 1/4 th of an hour passes between 2:24pm and 4:40pm
Between 2:24pm and 4:40pm 2.26 part of hours has been passed.
What is a fraction?A fraction is written in the form of p/q, where q ≠ 0.
Fractions are of two types they are proper fractions in which the numerator is smaller than the denominator and improper fractions where the numerator is greater than the denominator.
Given, What part of an hour passes between 2:24 pm and 4:40 pm.
Now (4 : 40 - 2 : 24 ) = 2 and 16 minutes.
16 minutes is 16/60 = 4/15 or 0.26 hours.
So, in total (2 + 0.26) = 2.26 part of an hour have been passed.
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how can we prepare for exam when 15 day are left
Answers
Answer:
1st step, DON'T PANIC. You have plenty of time to study. When it's a big exam, I like to set aside 2 hours daily for studying. If you're confused by certain topics, watch a video or ask your professor how to properly complete the answer. One college tip I learned: ALWAYS ASK THE PROFESSOR IF YOU GET CONFUSED WHILE STUDYING! They can help clear up problems you have. On the night before the exam, make sure you get a good night's rest. On the day of your exam, make sure to eat a good breakfast.
You have plenty of time, you got this! :)
1. Have some healthy food...or junk food or food that you love... because with a hungry stomach no one can study...
2. Take a break after every 15min... while studying..
3. Call your friends and study together...i.e. make a study circle...
4.Study loudly and faster...
At last..if you feel you're prepared then... good..if not.. then have a corner and weep and cry...or
5. have advice of your best friend or genius friend... if he or she didn't picks up phone then again cry because he or she is studying seriously with concentration so he or she will not pick up your phone...
6. At last pick up your all the methods of cheating...
hope it helps..
What is the solution to this inequality?
X-7>-6
O A. X<-13
O B. x>-13
O C.x>1
O D x< 1
Answers
Answer:
Option C
X>1
Step-by-step explanation:
x-7>-6
x>-6+7
x>1
Please Mark me brainliest
Answer:
C
Step-by-step explanation:
x - 7 > -6
x > -6 + 7
x > 1
• We change the inequality sign if we multiply or divide a negative number from both sides. But since it’s positive and we added it we won’t change the inequality sign.
find the value of the trigonometric ratio
Answers
tan c =opposite÷ adjacent
tan c =36÷15
tan c = 12÷5
Please answer this!!!
Answers
Answer:
12/13
Step-by-step explanation:
we know this is a right triangle, and we know that the hypotenuse is the longest side. so we can already say that the hypotenuse of the triangle is DF or 78.
when you draw it out, you find that the shortest leg is 30 and the second is 72 and the hypotenuse is 78. we also know that cosine is adjacent/hypotenuse. once you draw the triangle and notice which side is adjacent to angle F, just fit it into the formula and you get:
cos F = 72/78
cos F = 12/13
Answer: Choice C. [tex]\frac{12}{13}[/tex]
Step-by-step explanation:
It helps to draw a right triangle with hypotenuse DF=78, and adjacent EF=72.Plug in cosine adjacent over hypotenuse and simplify.5. Which expression is a simplified form of (3x2 + 2x - 4) + (-7x2 + 6x - 2)?
a 10x2 + 8x - 6
b. -4x2 + 8x - 2
C.-4x2 + 8x + 2
d. -4x2 + 8x - 6
Answers
Answer:
d. -4x² + 8x - 6
Step-by-step explanation:
(3x² + 2x - 4) + (-7x² + 6x - 2)
3x² + 2x - 4 - 7x² + 6x - 2
3x² - 7x² + 2x + 6x - 4 - 2
-4x² + 8x - 6
which of the statement is true
Answers
Answer:
c
Step-by-step explanation:
Profit = 4.5 (240)-1080 = 0
the maximum profit would occur when u sell all 350 tickets,
profit = 4.5(350)-1080 = 495 (not a nor b)
Answer:
The statement that is true is C
a microwave oven has a capacity of 1.59ft³. the interior of the microwave of the is 14 in wide and 14 in deep. what is the height of the interior of the microwave, to the nearest tenth of a foot
Answers
Step-by-step explanation:
The easiest way to do this is to imagine the microwave as a box. Ignore the turntables, whether or not something will spin if it touches an edge, and anything else that would make the fact that we're talking about a microwave confusing. This is a box now. :)
The capacity of our box is 1.59 ft^3, which means it can hold that much of "things" inside it. Instead of a capacity, we'll think of it as the box's total area. A tricky thing to note, is the area is given in ft, where the interior values are given as inches. So we'll need to turn the inches into feet.
To get the area of a box, we do length x width x height.
We're already given the length, 14 (inches), and the width, 14 (inches), so all we need to find now is the height (feet). To turn the lengths into feet, we'll divide them by 12, because there are 12 inches in 1 foot, or Feet = Inches / 12.
To finally turn it into math:
Area (A) = length (L) * width (W) * height (H)
[tex]A = L * W * H[/tex]
Insert known values.
[tex]1.59 = \frac{14}{12} * \frac{14}{12} * H[/tex]
Multiply.
[tex]1.59 = \frac{196H}{144}[/tex]
Simplify.
[tex]1.59 = \frac{49H}{36}[/tex]
Multiply both sides by 36.
[tex]1.59*36 = \frac{49H}{36}*36[/tex]
Simplify.
[tex]57.24 = 49H[/tex]
Divide both sides by 49.
[tex]\frac{57.24}{49} = \frac{49H}{49}[/tex]
Simplify.
[tex]1.16816326... = H[/tex]
To turn make this the nearest Tenth, we round up to the first decimal place.
Answer:
H = 1.2 ft
The hypotenuse of a right triangle is 52 in. One leg of the triangle is 8 in. more than twice the length of the other. What is the perimeter of the triangle?
20 in.
26 in.
120 in.
138 in.
Answers
Answer:
its c 120in
Step-by-step explanation:
The perimeter of the triangle is,
⇒ 138 in.
What is mean by Triangle?A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
The hypotenuse of a right triangle is 52 in.
And, One leg of the triangle is 8 in. more than twice the length of the other.
Hence, We get;
Lengths of legs are,
⇒ x
And, ⇒ 8 + 2x
Hence, We can formulate;
⇒ 52² = x² + (2x + 8)²
⇒ 2704 = x² + 4x² + 64 + 24x
⇒ 5x² + 24x - 2640 = 0
⇒ x = 20 and x = - 132/5
For perimeter;
Take x = 20
Hence, The perimeter of the triangle is,
⇒ 52 + x + (2x + 8)
⇒ 52 + 20 + (2 × 20 + 8)
⇒ 52 + 20 + 48
⇒ 138 in.
Thus, The perimeter of the triangle is,
⇒ 138 in.
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Can someone please help me with this I don't understand take your time if you can explain it
Answers
:)
Step By Step Explanation:
I"ll repost it for you
the train regularly travels 135 miles in 60 minutes.
What is the speed, in miles per minute, on this portion of the route?
Answers
Answer:
2.25 miles per minute
Step-by-step explanation:
Speed = Distance/Time
135/60
2.25 miles per minute
Answer from Gauthmath
TIMED PLEASE HELP EDGE 2021
Answers
RSM wants to send four of its 18 Math Challenge teachers to a conference. How many combinations of four teachers include exactly one of either Mrs. Vera or Mr. Jan?
Answers
Answer:
1120 combinations of four teachers include exactly one of either Mrs. Vera or Mr. Jan.
Step-by-step explanation:
The order in which the teachers are chosen is not important, which means that the combinations formula is used to solve this question.
Combinations formula:
[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this question:
1 from a set of 2(Either Mrs. Vera or Mr. Jan).
3 from a set of 18 - 2 = 16. So
[tex]C_{2,1}C_{16,3} = \frac{2!}{1!1!} \times \frac{16!}{3!13!} = 1120[/tex]
1120 combinations of four teachers include exactly one of either Mrs. Vera or Mr. Jan.
draw the graph of the line: with a slope of -1/2 and that passes through the origin
Answers
Answer:
I hope this helps!
HELP ME ASAP!!!
two angles are supplementary. one angle is three less than 4 times it's supplement. what are the measures of the two angles?
Answers
Step-by-step explanation:
Two angles are supplementary one angle is 4 degrees less than three times the other find the measures of the angles
x + y = 180
x = 3y - 4
3y - 4 + y = 180
4y = 184
y = 46
x = 134
Answer:
30°，150°
Step-by-step explanation:
30 points PLEASE HELP!!!!
Find the volume of the pyramid.
76.34 ft3
458.06 ft3
916.11 ft3
305.37 ft3
Answers
Answer:
916.11 ft³
Step-by-step explanation:
(9×6)×7.83/2×13/3
= 211.41×13/3
= 916.11 ft³
Answered by GAUTHMATH
Answer:
916.11 ft³
Step-by-step explanation:
A tax on which of these products or services would not be considered a sin
tax"?
O A. Alcohol
O B. Gambling
O C. Electronics
O D. Tobacco
Answers
Answer: electronics
Hope that helps
The option that is not considered a sin tax is option B, gambling.
What is a sin tax?
A sin tax is a tax that is applied to "harmful things" to society or individuals. An example of this is Tobacco.
From the given options, the harmful ones are alcohol, gambling, and tobacco. These 3 are considered harmful, so would be included under the sin taxes.
Then the option that is not considered a sin tax is option B, gambling.
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