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
Instructions: Find the surface area of each figure. Round your answers to the nearest
tenth, if necessary.
Answers
Answer:
136
Step-by-step explanation:
Surface area=2*(lb+bh+lh)
=2*(12+42+14)=2*68=136
Solve simultaneous equations 3x-2y=17 +2x-2y=10
Answers
Answer:
(7;2)
Step-by-step explanation:
[tex]3x - 2y = 17 \\ - 2y = 17 - 3x \\ 2x + 17 - 3x = 10 \\ 2x - 3x = 10 - 17 \\ - x = - 7 \\ x = 7 \\ - 2y = 17 - 21 \\ - 2y = - 4 \\ y = 2[/tex]
9) Assume that the random variable X is normally distributed, with mean = 90 and standard deviation o = 12. Compute the probability P(57 < X < 105).
A) 0.7888 B) 0.8944 C) 0.8914 D) 0.8819
Answers
Answer:
Step-by-step explanation:
The mean, [tex]\bar{x}[/tex], is 90 and the standard deviation, [tex]\sigma[/tex], is 12. We are looking for the probability that the variable X will fall between 57 and 105. We use the z-score table for this, AFTER we find the z scores. The formula to find the z-scores for us is:
[tex]P(\frac{57-\bar{x}}{\sigma}\leq z\leq \frac{105-\bar{x}}{\sigma})[/tex] and we fill in accordingly:
[tex]P(\frac{57-90}{12}\leq z\leq \frac{105-90}{12})[/tex] which simplifies to
[tex]P(-2.75\leq z\leq 1.25)[/tex] and we will break them up into 2 different sets as follows:
P(-2.75 ≤ z ≤ 0) + P(0 ≤ z ≤ 1.25)
and based on the fact that z scores are given from 0 on up, we are going to convert the first one by using the logic that if z is greater than -2.75 but less than 0, by symmetry, z is greater than 0 but less than 2.75:
P(0 ≤ z ≤ 2.75) + P(0 ≤ z ≤ 1.25) and we go to the z-score table.
Locate 2.7 down along the left side and move over til you're under the .05; that gives us the z-score for 2.75 which is .4970. Do the same for 1.25 to get a z-score of .3944. Add them together to get a final z-score that covers the range of values for X:
.4970 + .3944 = 0.8914
please find the value of x
Answers
Answer:
x = 5.9
Step-by-step explanation:
Since this is a right triangle, we can use trig functions
tan theta = opp /adj
tan 26 = x /12
12 tan 26 =x
x=5.85279
Rounding to the nearest tenth
x = 5.9
with work please and thanks
Answers
Answer:
Step-by-step explanation:
He bought a total of 5 tickets, some child, some adult. This is the NUMBER equation.
# of adult tickets + # of child tickets = 5 tickets
Now for the COST equation. Each child ticket costs $2, so the expression for that is 2c; each adult ticket costs $8, so the expression for that is 8a. He spend a total of $22:
$2c + $8a = $22
Money and the number of tickets are NOT THE SAME so they cannot be put into the same equation. They won't add or subtract because they're not like.
Your system is choice A
Roger bought some tins and decided to fill them with brownies to give to his friends. Roger baked 278 brownies. He put 4 brownies in each tin and made sure to fill as many tins as he could. How many brownies did Roger have left over?
Answers
Answer:
Divide 278 by 4 and the remainder value is the amount of leftover brownies.
278 ÷ 4 = 69, remainder of 2
Therefore, he had 2 brownies left.
6. 792 = *
A. (7 x 10) + (3 x 4) + (2 x 1)
B. (7 x 11) + (3 x 3) + (2 x 1)
C. (7 x 11) + (3 x 4) + (2 x 1)
D. (7 x 10) + (3 x 3) + (2 x 1)
Answers
Answer:
D. (7×10) + (3×3) + (2×1)
Step-by-step explanation:
(7×10)=70 + (3×3)=9 + (2×1)=2
=70+9+2
=792
Question No.b
Find the products or quotient in the exponential forms by using law of indices
The answer will be :- 3^9
Answers
Answer:
3*6 × 3*-3 × 3*6
3*(6-3+6)
3*9
Find the length of CV
Answers
Answer:
sinΘ = opposite/ hypotenuse
sin (37)= CV/55
55 sin (37) = CV
CV=33.1
OAmalOHopeO
Joshua buys a $6000 car on the following terms:
30% deposit
• monthly repayments of $210 for 2 years.
Find:
a) the deposit paid.
Answers
pls help me i really need it!
Answers
Answer:
A
Step-by-step explanation:
Angles A and B are complementary. The measure of angle A
is yº. What is the measure of angle B?
Answers
Answer:
D, [tex](90-y)[/tex] degrees
Step-by-step explanation:
Complementary angles are angles that add up to 90 degrees. Since angle A and angle B are complementary, they add up to 90. Since the value of angle A is y degrees, we can substitute y into the equation to get
[tex]y+B=90[/tex]
To get the measure of angle B, we must isolate B.
To isolate B, we have to remove every term from its side.
The only term on the side that B is on other than B is y. To remove y from the left side, we must subtract y from both sides. Doing this will result in the equation:
[tex]B=90-y[/tex]
The only answer option that meets this criteria is D.
4over6=24over16 explain the error in the students work
Answers
Step-by-step explanation:
The key to solve this problem is using ratios and proportions.
The key to solve this problem is using ratios and proportions.Ratio is the relationship between two numbers, defined as the quotient of one number for the other. So: The ratio between two numbers a and b is the fraction a/b and it is read a to b. This reason can also be written a : b.
The key to solve this problem is using ratios and proportions.Ratio is the relationship between two numbers, defined as the quotient of one number for the other. So: The ratio between two numbers a and b is the fraction a/b and it is read a to b. This reason can also be written a : b.Given two reasons a/b and c/d we say that they are in proportion if a/b = c/d. The terms a and d are called extremes while b and c are the means. In every proportion the product of the extremes is equal to the product of the means: a.d = b.c
The key to solve this problem is using ratios and proportions.Ratio is the relationship between two numbers, defined as the quotient of one number for the other. So: The ratio between two numbers a and b is the fraction a/b and it is read a to b. This reason can also be written a : b.Given two reasons a/b and c/d we say that they are in proportion if a/b = c/d. The terms a and d are called extremes while b and c are the means. In every proportion the product of the extremes is equal to the product of the means: a.d = b.cA student uses the ratio of 4 oranges to 6 fluid ounces of juice to find the numbers of oranges needed to make 24 fluid ounces of juice.
The key to solve this problem is using ratios and proportions.Ratio is the relationship between two numbers, defined as the quotient of one number for the other. So: The ratio between two numbers a and b is the fraction a/b and it is read a to b. This reason can also be written a : b.Given two reasons a/b and c/d we say that they are in proportion if a/b = c/d. The terms a and d are called extremes while b and c are the means. In every proportion the product of the extremes is equal to the product of the means: a.d = b.cA student uses the ratio of 4 oranges to 6 fluid ounces of juice to find the numbers of oranges needed to make 24 fluid ounces of juice.
The key to solve this problem is using ratios and proportions.Ratio is the relationship between two numbers, defined as the quotient of one number for the other. So: The ratio between two numbers a and b is the fraction a/b and it is read a to b. This reason can also be written a : b.Given two reasons a/b and c/d we say that they are in proportion if a/b = c/d. The terms a and d are called extremes while b and c are the means. In every proportion the product of the extremes is equal to the product of the means: a.d = b.cA student uses the ratio of 4 oranges to 6 fluid ounces of juice to find the numbers of oranges needed to make 24 fluid ounces of juice. The error in the student's work was that they reversed the reason, 24/16 instead of 16/24.
Find the measure of one exterior angle for the following regular polygon.
Answers
Answer:
40 degrees
Step-by-step explanation:
The formula for calculating the sum of interior angles is ( n − 2 ) × 180 (n = Number of sides)
in this scenario you have 9 sides so put it in the equation.
(9 - 2) ×180 = 1260
1260 is the sum of all the interior angles so we must divide this number by the number of sides to get the angle of one individual interior angle.
1260/9 = 140
180-140 = 40 degrees
What are the roots of this equation x^2-4x+9=0?
Answers
Answer:
Roots is a fancy word for zero, which is also another fancy word for the x intercept.
You can factor a quad or use the quadratic equation to get zeros.
In this case, its non-factorable. So you gotta use the quadratic.
Using the quadratic equation you get : x = -2 ± √-1 (√5),
which can be rewritten as x = -2 ± i(√5)
Function was : x^2 + 4x + 9
For quad, I identified the following:
A = 1
B = 4
C = 9
I also added a picture of the formula in the attached.
how do i solve this??
Answers
Answer:
10q^2+3q
Step-by-step explanation:
All you have to do is combine like terms I think
(I haven't done this stuff in ages so I'm most likely wrong)
Answer:
10q^2+3q
Step-by-step explanation:
2q^2 +3q +8q^2
Combine like terms
10q^2+3q
Find the volume of this cylinder.
As soon as possible please..
Answers
Answer:
144
Step-by-step explanation:
V=PIr^2h
PI=3
V=3r^2h
r=2 h=12
V=3(2^2)12
V=3(4)12
V=12(12)
V=144in^3
Step-by-step explanation:
36 in ^3 is the volume of cylinder
what is -3/5 as a decimal
Answers
Answer:
-0.6
Step-by-step explanation:
Answer:
⅗ as a decimal is -0.6
Hope it's right if not then sorry, have a great day:)ANSWER QUICK! MANY POINTS! NO WRONG ANSWERS PLEASE!
1. Use the table and the graph to answer the questions.
Function 1
x: -1 -2 -3 2 3
y: 3 5 7 -3 -5
Function 2
(see picture below)
a) What is the rate of change for each function? Show your work.
b) Which function has the greater rate of change?
Answers
Answer:
see below
Step-by-step explanation:
Function 1
Find the slope
m = ( y2-y1)/ (x2-x1)
= (-5 -3) /(3 - -1)
= ( -5-3)/(3+1)
= -8/4
= -2
Function 2
Using the points (0,4) and (-1,0)
m = ( y2-y1)/ (x2-x1)
m = ( 0-4)/(-1-0)
= -4/-1
= 4
The function with the greater rate of change is function 2
#Function 1
(-1,3)(-2,5)Slope=
m=5-3/-2+1m=2/-1m=-2#Function 2
(-1,0)(0,4)Slope
m=4/1m=4Function 2 has greater rate of change
Use the function to answer questions #5-7.
please help with both
Answers
Answer:
2lnf .÷f35%vbbcy^hjv6bs8nc
Evaluate the functions in the given points.
a. f(x) = x
2 − 6 for f(0), f(−6), f(√6), f(
1
2
)
b. f(x) = x
3 + 2x for f(0), f(−2), f(−1), f(
1
2
)
c. f(x) =
1−2x
3
for f(0), f(−2), f(2), f (
1
2
) , f(m), f(m − 1), f(−m)
d.f(x) = |x| −
1
x
for f(−1), f(2), f(h), f (
1
2
) , f(−h + 2) where h ≠ 0
Answers
Answer:
Hey I'm sorry I didn't get to answer your question it's just that I need the points because I don't have enough to get help with my question. I hope you get the answer that you need for you question. Good Luck :)..............
Step-by-step explanation:
find the area of the right triangle
Answers
Answer:
the area of right angle is 1.5×2.5=3.75
Answer:
1.5 units^2
Step-by-step explanation:
If we say that the missing side is x, we can use the pythatgorean theorem to find it.
1.5^2 + x^2 = 2.5^2 => 2.25 + x^2 = 6.25 => x^2 = 4 => x=2
The area of a triangle is hieght * base * 1/2 => 1.5 * 2 * 1/2 = 1.5
Someone help me please
Answers
Answer:
16 cows
Step-by-step explanation:
i just guessed and checked using
4*cow heads + 2 * geese heads
i got 16 cows and 47 geese
Which represents a unit rate?
Answers
Answer:
A unit rate is a rate with 1 in the denominator. If you have a rate, such as price per some number of items, and the quantity in the denominator is not 1, you can calculate unit rate or price per unit by completing the division operation: numerator divided by denominator.
.Prove a+ar+ar2+………+arn-1=a(rn -1 ) r-1
Answers
Answer:
See below.
Step-by-step explanation:
Call the sum of the first n terms S.
S = a + ar + ar^2 + ar^3 + ... + ar^(n -1)
Multiply both sides by r.
Sr = ar + ar^2 + ar^3 + ar^4 + ... + ar^n
No subtract S - Sr.
S = a + ar + ar^2 + ar^3 + ar^4 + ... + ar^(n -1)
(-) Sr = ar + ar^2 + ar^3 + ar^4 + ... + ar^(n - 1) + ar^n
-------------------------------------------------------------------
S - Sr = a - ar^n
S(1 - r) = a(1 - r^n)
S = a(1 - r^n)/(1 - r)
S = a(r^n - 1)/(r - 1)
a + ar + ar^2 + ar^3 + ... + ar^(n -1) = a(r^n - 1)/(r - 1)
Multiply the polynomials.
Answers
[tex]\\ \sf\longmapsto( 7x {}^{2} + 9x + 7)(9x - 4) \\ \\ \sf\longmapsto {7x}^{2} (9x - 4) + 9x(9x - 4) + 7(9x - 4) \\ \\ \sf\longmapsto 63x {}^{3} - 28 {x}^{2} + {81x}^{2} - 36x + 63x - 28 \\ \\ \sf\longmapsto {63x}^{3} + 53 {x}^{2} + 27x - 28[/tex]
Option d is correct
hipe it helps you...............
hipe it helps you...............
What does x equal in the following equation 3x + 5 = 17 A x = 2 B x = 4 C x = 5
Answers
Answer:
see below
Step-by-step explanation:
3x + 5 = 17
Subtract 5 from each side
3x+5-5 = 17-5
3x = 12
Divide each side by 3
3x/3 =12/3
x =4
Answer:
B (x=4)
Step-by-step explanation:
plug it in...
3(4)+5=17
What number divided by 8 equals 11?
Equation:
Solution :
Answers
Answer:
equation: x/8 = 11
solution: 88
Step-by-step explanation:
Let our unknown number be the variable, x. Since we're dividing x by 8, we can rewrite it into these expressions: x/8 or x ÷ 8. Because x divided by eight equals eleven, we can translate it into this equation: x/8 = 11, or x ÷ 8 = 11.
To solve for the equation x/8 = 11, let's multiply each side by 8 so we can isolate the variable, x.
x/8 ⋅ 8 = 11 ⋅ 8
x = 88
Now that we've completed multiplying each side by 8, you can see that x = 88, meaning that the unknown number is 88.
Answer:
88
Step-by-step explanation:
Equation: 88/8
Solution: 11
The surface area of a cylinder is given by the formula
S = 30πr + 2πr2
How many times greater is this compared to a circle with A = πr2
Answers
Answer:
[tex] x = \frac {30}{r} + 2 [/tex]
Step-by-step explanation:
Given the following data;
S.A of cylinder = 30πr + 2πr²Area of circle = πr²To find how many times greater is the S.A compared to the area, we would have to divide the surface area (S.A) of the cylinder by the area of circle.
Let the unknown variable be x.
[tex] x = \frac {30 \pi r + 2 \pi r^{2}}{\pi r^{2}} [/tex]
Factorizing the numerator, we have;
[tex] x = \frac {\pi r(30 + 2r)}{\pi r^{2}} [/tex]
Dividing both sides by πr, we have;
[tex] x = \frac {30 + 2r}{r} [/tex]
Simplifying further, we would have;
[tex] x = \frac {30}{r} + 2 [/tex]
What is the first step in solving this system by elimination?
2x + 10y = -20
-x + 4y = 28
A. Multiply the first equation by -2.
B. Combine the equations as they are.
C. Multiply the second equation by 2.
D. Multiply the second equation by -4.
Answers
You want to try and get rid of one of the variables ( x or y) so you would multiply the second equation by 2 which would make the x variable -2x which then when added to the first equation would eliminate the x variable.
Answer: c. Multiply the second equation by 2.
