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if z1= 3+3i and z2=7(cos(5pi/9) + i sin (5pi/9)), then z1/z2= blank
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[tex]z1=\stackrel{a}{3}+\stackrel{b}{3}i~~ \begin{cases} r = \sqrt{a^2+b^2}\\ r = \sqrt{18}\\[-0.5em] \hrulefill\\ \theta =\tan^{-1}\left( \frac{b}{a} \right)\\ \theta =\frac{\pi }{4} \end{cases}~\hfill z1=\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right] \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}\left[\cos\left( \frac{\pi }{4} \right) i\sin\left( \frac{\pi }{4} \right) \right]} {7\left[\cos\left( \frac{5\pi }{9} \right) i\sin\left( \frac{5\pi }{9} \right) \right]} \\\\[-0.35em] ~\dotfill\\\\ \qquad \textit{division of two complex numbers} \\\\ \cfrac{r_1[\cos(\alpha)+i\sin(\alpha)]}{r_2[\cos(\beta)+i\sin(\beta)]}\implies \cfrac{r_1}{r_2}[\cos(\alpha - \beta)+i\sin(\alpha - \beta)] \\\\[-0.35em] ~\dotfill[/tex]
[tex]\cfrac{z1}{z2}\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{\pi }{4}-\frac{5\pi }{9} \right)+i\sin\left( \frac{\pi }{4}-\frac{5\pi }{9} \right) \right] \\\\\\ \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{-11\pi }{36} \right) +i\sin\left( \frac{-11\pi }{36} \right) \right]\implies \cfrac{\sqrt{18}}{7}\left[\cos\left( \frac{83\pi }{36} \right) +i\sin\left( \frac{83\pi }{36} \right) \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill \cfrac{z1}{z2}\approx 0.348~~ + ~~0.496i~\hfill[/tex]
The value of z1/z2 is √18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).
What is complex number?"A complex number is the sum of a real number and an imaginary number and it is of the form x + iy and is usually represented by z".
For the given situation,
z1= 3+3i and
z2= 7(cos(5π/9) + i sin (5π/9))
To divide the complex numbers, both should be in same form.
Convert z1 in polar form.
z is of the form x+iy, so, r=[tex]\sqrt{x^{2}+y^{2} }[/tex]
⇒[tex]r=\sqrt{3^{2}+3^{2} }[/tex]
⇒[tex]r=\sqrt{18}[/tex]
θ = [tex]tan^{-1}(\frac{b}{a} )[/tex]
⇒[tex]tan^{-1}(\frac{3}{3} )[/tex]
⇒[tex]tan^{-1}(1)[/tex]
⇒[tex]45[/tex]°
The polar form is of the form, z= r (cosθ + i sinθ),
⇒ z1 = [tex]\sqrt{18}[/tex] (cosπ/4 + isinπ/4)
The formula for dividing complex number is
z1/z2 = r1(cos θ1 + isin θ1) / r2(cos θ2 + isin θ2)
⇒ z1/z2 = r(cosθ + isinθ)
where, r = r1/r2 and θ = (θ1 - θ2)
z1/z2 = [tex]\sqrt{18}[/tex] (cos π/4 + isin π/4) / 7 (cos 5π/9 + isin 5π/9)
⇒ r = [tex]\sqrt{18}[/tex] / 7 and
θ = (π/4 - 5π/9 )
⇒ θ = (-11π/36)
cos(-θ) = cos θ
⇒cos( -11π/36 ) = cos ( 11π/36 )
sin(-θ) = -sin θ
⇒ sin ( -11π/36 ) = -sin ( 11π/36 )
Thus, z1/z2 = [tex]\sqrt{18}[/tex] / 7 (cos ( 11π/36 ) - isin ( 11π/36 ))
Hence we can conclude that the value of z1/z2 is
√18/ 7 (cos ( 11π/36 ) - isin ( 11π/36 )).
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Find the equation of the line passing through the point (−5,−9) and parallel to the line y=3x+2 .
Answers
Answer:
Root:(-⅗,0)
vertical intercept:(0,5)
Step-by-step explanation:
A movie theater charges $8.00 for adults and $5.00 for children. If there were 40 people altogether and the theater collected $272.00 at the end of the day, how many
of them were adults?
A. 29 adults
B. 16 adults
C. 24 adults
D.10 adults
Answers
Answer:
C.24
Step-by-step explanation:
First multiply 8 by 24 this is so you know how much 24 adults would be. You will get 192 so next you need to subtract 192 from the total money 272 to get 80 and 80 should be how much all of the kids cost now to check and make sure there are 40 people divide 80 by 5 to get 16 which is the total number of kids no just add 16 and 24 together to get 40
Evaluate. 5.8 – 3.63 = ________
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I Hope This Helps
the line passes through the point (4,7) and has a slope that is reciprocal of the slope of the line y=1/3x+2
point slope form and slope intercept form pls
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first you have to change 1/3 to -3
y=-3x+2
then you have to replace 2 with 'b' because you don't know the y intercept yet
y=-3x+b
then you substitute the point (4,7) into the equation
7= -3(4) + b
7= -12 + b
19 = b
y = -3x + 19 is the equation in slope intercept form
for point slope form:
-3x is still the slope and you replace x and y with the point 4,7
y- 7= -3 (x-4)
Find the digits in the tens place, in the hundredths place, and in the thousandths place for the following number 748.6291
Answers
Answer:
8 for tens, 9 for hundredths and thousandths is 1
Step-by-step explanation:
you can find the number by looking at the place of the numbers if it is tens look at the tens place which is the number eight as tens are 2 digit numbers
a. How much higher is 500 than 400 m?
b. How much higher is 500 than -400 m?
C. What is the change in elevation from 8,500 m to 3,400 m?
d. What is the change in elevation between 8,500 m and -300 m?
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B 900m
C -5,100m
D- 8,800
I need to know the area of this triangle and I would like to have an explanation because I dont know how to do it I know the formula just not how to get the height
Answers
Answer:
18 square units
Step-by-step explanation:
When one side is vertical or horizontal, it is convenient to find the height from the opposite vertex to that side. It is generally an integer number of grid squares.
Here, the height is 4 -(-2) = 6. The base length is 1 -(-5) = 6. That means the area is ...
A = 1/2bh
A = 1/2(6)(6) = 18 . . . . square units
_____
Alternate solution
If you want to go at this the hard way, you can find the length of LK and the distance of J to that line.
The length of LK is ...
d = √((x2 -x1)^2 +(y2 -y1)^2) = √((4 -(-2))^2 +(3 -(-5))^2) = √(36 +64) = 10
The distance to line LK can be found by writing the equation of the line LK in general form:
4x -3y -7 = 0
Then the distance formula for point (x, y) is ...
d = |4x -3y -7|/√(4^2 +3^2)
d = |4(-2) -3(1) -7|/5 = |-8 -3 -7|/5 = 3.6
Using the area formula with these dimensions gives ...
A = 1/2bh = 1/2(10)(3.6) = 18 . . . . square units
Answer:
18 square units
Step-by-step explanation:
pls pls pls mark me as brainliest
GUYS PLEASE HELP ME ON THIS ONE!, i will give brainliest to the right answers
Aisha has $40 to spend for her ornithology club. She spends some of it buying birdseed and saves the rest. Her savings is given by f(x)=40−1.25x, where x is the number of pounds of birdseed she buys at $1.25 per pound.
Answers
Answer:
Step-by-step explanation:
Given f(x)=40-1.25x, where x=pounds of birdseed and 40=her saved amount, and birdseed costs 1.25/lb.
f(3)=40-1.25(3)
f(3)40-3.75
f(3)=36.25
If she buys (3 pounds) of birdseed, she saves ($36.25).
f(18)=40-1.25(18)
f(18)=40-22.5
f(18)=17.5
If she buys (18 pounds) of birdseed, she saves ($17.5)
f(36)=40-1.25(36)
f(36)=40-45
f(36)=-5
If she buys (36 pounds) of birdseed, she will need an extra $5 to pay off her debt.
If she buys (36 pounds) of birdseed, she will need an extra $5 to pay off her debt.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
Given function f(x)=40-1.25x,
where x=pounds of birdseed and 40 is her saved amount, and birdseed costs is 1.25/lb.
f(3)=40-1.25(3)
f(3)40-3.75
f(3)=36.25
If she buys (3 pounds) of birdseed, she saves ($36.25) then;
f(18)=40-1.25(18)
f(18)=40-22.5
f(18)=17.5
If she buys (18 pounds) of birdseed, she saves ($17.5) then;
f(36)=40-1.25(36)
f(36)=40-45
f(36)=-5
If she buys (36 pounds) of birdseed, she will need an extra $5 to pay off her debt.
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Solve 3 - x= -9.
O A. x= 6
O B. x= 12
O c. x= -6
O D. x=-12
Answers
Answer:
x=12
Step-by-step explanation:
you need the x alone so you substract 3 from both sides.
Then you have, -x= -12
You need the x positive so you change the sign for it but as it is an equation you need to do the same thing on the other side, making both sides positive.
x=12
-x + 3 = -9
Subtract 3 from both sides of the equation. Simplify
log x √x + log √c c² = ?
[tex] log_{x}\sqrt{x} + log_{ \sqrt{c} } {c}^{2} = \\ [/tex]
please help me!!!!?;
Answers
Given that
log √x (x) + log c² (c)
⇛ log x^1/2 (x) + log c² (c)
We know that
log a^m (b^n) = m/n log a (b)
⇛ 1/2 log x (x) + 2 log c (c)
We know that
log a ( a) = 1
⇛ (1/2)(1) + 2(1)
⇛ (1/2)+2
⇛ (1+4)/2
⇛ 5/2
Answer:- log √x (x) + log c² (c) = 5/2
[(x) ,(c) , (a) represents the bases ]
Additional comment:
(a+b)(a-b) = a²-b²log a^m (b^n) = m/n log a (b)log a ( a) = 1also read similar questions: if log^2(x)=log^2(a)+log^2(b)-log^2(c) then x is....
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simplify the following expression. log(x2) - log(x) A. log(x^2+x) B. log(x^3) C. log(x^2-x) D. log(x)....
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Integrate:Cos 7x/(1+sin**2(7x))
Answers
Answer:
7x/1+7sin(2)x
Step-by-step explanation:
Line f is parallel to line g.which statements about line f and g is true
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Answer:
Step-by-step explanation:
They have different y intercepts.
They have the same slope.
Here are 2 examples
y = 3x + 5
y = 3x + 19
The slopes are both three. The y intercepts are 5 for the top equation and 19 for the bottom equation.
They are both lines. That means their domain is -infinity <x < + infinity
Their range is - infinity <y < + infinity.
We are excluding examples like x = 3 and x = 5 and y = 1 and y = 3.
It is true that they both are parallel in both cases, but the discussion for each pair is a little different.
What is the percent of decrease from 80 to 68?
Answers
Answer:
-15%.
Step-by-step explanation:
Subtract 68 by 80 to get -12.
Divide -12 by the absolute value of 80 to get -0.15.
Multiply 100 to -0.15 to get -15.
A box is filled with five green card six green card and three blue cars. A card is chosen at random from the box. What is the Probability that it is a green or blue card
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Answer:
so there is 14 cards in total so out of those 14 14 cards are either blue or green so 100%
Hope This Helps!!!
On the 1st January 2014 Carol invested some money in a bank account.
The account pays 2.5% compound interest per year.
On 1st January 2015 Carol withdrew £1000 from the account.
On 1st January 2016 she had £23517.60 in the account.
Work out how much Carol originally invested in the account.
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Answer:
didn't know how to workout iam only 3rd standard
Answer:
Step-by-step explanation:
bubble gum
Select all that apply.
Zack is taking a test in which he answers questions at a rate of question per minute. If Q represents questions and
M represents minutes, which of the following equations describe this rate relationship?
M= 20 x Q
Q = 2 x M
M=1/2 x Q
Q=1/2 x M
Answers
The equation Q = 2 x M represents the relationship between question and time.
What is an equation?An equation is a combination of different variables, in which two mathematical expressions are equal to each other.
Given that,
Zack takes a test, in which he answers at rate of 2 question per minute.
Also, Q represents questions
And M represents minutes.
According to given condition,
In 1 minute, Zack answers 2 question,
In M minutes, Zack answers 2 x M question,
Total questions presented by Q,
Q = 2 x M.
The required relationship for M and Q is Q = 2 x M.
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Omar deposited $4000 into an account with 4.4% interest, compounded quarterly. Assuming that no withdrawals are made, how much will he have in the account after 10 years?
Do not round any intermediate computations, and round your answer to the nearest cent.
Answers
future value
principal
int
compound
years
A retailer needs to purchase 11 printers. The first printer costs $58, and each additional printer costs 5% less than the price of the previous printer, up to 15 printers. What is the total cost of 11 printers?
$381.99
$500.19
$606.10
$609.00
Answers
Answer:
$500.19 is the answer
Step-by-step explanation:
pls vote branliest
The total cost of 11 printers is $500.19.
Given that,
A retailer needs to purchase 11 printers.
The first printer costs $58, and each additional printer costs 5% less than the price of the previous printer, up to 15 printers.
We have to determine,
What is the total cost of 11 printers?
According to the question,
The first printer costs $58.
Each additional printer costs 5% less than the price of the previous printer, up to 15 printers.
The cost of a second printer is,
= 0.95 × 58 = $55.1
The cost of a third printer is,
= 0.95 × 55.1 = $52.345
The cost of a fourth printer is,
= 0.95 × 52.345 = $49.727
The cost of a fifth printer is,
= 0.95 × 49.727 = $47.241
The cost of a sixth printer is,
= 0.95 × 47.241 = $44.879
The cost of a seventh printer is,
= 0.95 × 44.879 = $42.635
The cost of an eight printer is,
= 0.95 × 42.635 = $40.50
The cost of a nineth printer is,
= 0.95 × 40.50 = $38.478
The cost of tenth printer is,
= 0.95 × 38.478 = $36.554
The cost of the eleventh printer is,
= 0.95 × 36.554 = $34.726
Therefore,
The total cost of 11 printers is,
= $58 + $55.1 + $52.345 + $49.727 + $47.241 + $44.879 + $42.635 + $40.50 + $38.478 + $36.554 + $34.726 = $500.19.
Hence, The total cost of 11 printers is $500.19.
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-2 2/3 divided by 1 5/6 + 2
NO MIXED NUMBERS
Answers
Answer:
14/15
Step-by-step explanation:
Answer:
6/11
Step-by-step explanation:
-2 2/3 divided by 1 5/6 = -16/11 and when you add two to that you get 6/11.
pls help. Have tried this problem a couple times but my answers didnt work
Answers
Answer:
infinite solutions...
Step-by-step explanation:
Is there more information to this problem? It can be pretty much anything.
N=3M
Nina can be 30 and Maryna 10, Nina can be 3 and Maryna 1, Nina can be 15 and Maryna 5...
Answer:
N - 4 = 3(M - 4). Simplified further, N - 4 = 3(M - 4)
Given,
Nina is N years old now and;
Maryna is M years old now
If four years ago, Nina was three times as old as Maryna was, then
Four years ago, Nina's age was N - 4
Four years ago, Maryna's age was M - 4
Given that Nina was three times as old as Maryna was four years ago, it means that
N - 4 = 3(M - 4)
To solve further
N = 3m - 12 + 4
N = 3M - 8
Step-by-step explanation:
Apply distributive 15+22
Answers
Answer:37
Step-by-step explanation:
If angle CAB= x+40, angle ACB=3x+10, angle CBD= 6x, what is angle CAB
Answers
I’m pretty sure it’s B
Explanation:
Add 3x+10 + x+40 and set it equal to 6x. then when you solve for x you get 25
Hope this helps!
HELP!! I'm not sure what goes in the blank.
Answers
Answer:
10[tex]sin^{2} ([/tex]β[tex])[/tex]
Step-by-step explanation:
We can find this two ways, first by seeing in the step after it, cosines are canceled out. Since you already have 10[tex]sin^{2} ([/tex]β[tex])[/tex] on the next step, you can assume that (since only the cosines changed and the cosine next ot the blank was removed), the value is 10[tex]sin^{2} ([/tex]β[tex])[/tex].
You can also use double angle formulas from the previous step:
(sin(2β) = 2 sin(β) cos(β))and find that:
5 sin (2β) sin(β) = 5 * (2 sin(β) cos(β)) sin(β)) = (10 sin(β) sin(β)) cos(β) =
10[tex]sin^{2} ([/tex]β[tex])[/tex] cos(β)
But since cos(β) is already present, we can see that the answer is 10[tex]sin^{2} ([/tex]β[tex])[/tex]
Line j has an equation of y = 1/5x + 7. Perpendicular to line j is line k, which passes through
the point (-1, -3). What is the equation of line k?
Write the equation in slope-intercept form. Write the numbers in the equation as simplified
proper fractions, improper fractions, or integers.
Answers
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the equation above
[tex]\begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}\qquad \impliedby y=\stackrel{\stackrel{m}{\downarrow }}{\cfrac{1}{5}}x+7 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{\cfrac{1}{5}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{5}{1}}\qquad \stackrel{negative~reciprocal}{-\cfrac{5}{1}\implies -5}}[/tex]
so then line K has a slope of -5 and pass through (-1, -3)
[tex](\stackrel{x_1}{-1}~,~\stackrel{y_1}{-3})\qquad \qquad slope=m=-5 \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-3)}=\stackrel{m}{-5}[x-\stackrel{x_1}{(-1)}] \\\\\\ y+3=-5(x+1)\implies y+3=-5x-5\implies y=-5x-8[/tex]
an expression that is equivalent to 10x - 8 +0.5(2x - 3)?
-
Answers
Answer:
......................
please help!! easy but i don't understand <3
Answers
in essence it's just asking on getting the equation of that line in more or less slope-intercept form, hmmm well, it's a line, so all we need is two points to get it, hmmm let's see let's use the starting point and terminal point, so we know it goes through the origin, (0,0) and also goes through (1, 0.7), let's change 0.7 to a fraction, so its 7/10, ok, let's use those two points from it.
[tex](\stackrel{x_1}{0}~,~\stackrel{y_1}{0})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{\frac{7}{10}}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{\frac{7}{10}}-\stackrel{y1}{0}}}{\underset{run} {\underset{x_2}{1}-\underset{x_1}{0}}}\implies \cfrac{~~ \frac{7}{10}~~}{1}\implies \cfrac{7}{10}[/tex]
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{0}=\stackrel{m}{\cfrac{7}{10}}(x-\stackrel{x_1}{0})\implies y = \cfrac{7}{10}x[/tex]
PLEASE ANSWER FOR 30 POINTS
Answers
Answer:
One Solution
Step-by-step explanation:
x + 3y = 3
x - y = -3
4y = 6 Subtract to solve by elimination.
y = [tex]\frac{3}{2}[/tex] Divide by 4.
x - [tex]\frac{3}{2}[/tex] = -3 Substitute [tex]\frac{3}{2}[/tex] for y in the original equation.
x = -[tex]\frac{3}{2}[/tex] Add [tex]\frac{3}{2}[/tex].
From a sample of 45 students exam scores, it was found that there was a mean score was 75 and a standard deviation of 5. Assume the distribution of exam scores is normal.
construct and interpret a 95% confidence interval for the true mean score.
Answers
Answer:
The 95% confidence interval for the true mean score is [tex][73.539,76.461][/tex]
Step-by-step explanation:
Note that [tex]CI=\bar x\pm z\frac{s}{\sqrt{n}}[/tex] where [tex]\bar x[/tex] is the sample mean, [tex]z[/tex] is the upper critical value for the desired confidence level, [tex]s[/tex] is the sample standard deviation, and [tex]n[/tex] is the sample size.
Therefore, [tex]CI=75\pm 1.96\frac{5}{\sqrt{45}}\approx[73.539,76.461][/tex]
Evaluate the expression 5xy+ -2xy+y^2 when x=4 and y=-5
Answers
Answer:
-35
Step-by-step explanation:
5(4)(-5) + -2(4)(-5) + (-5)²
= -100 + 40 + 25
= -35
Answer: -35
Step-by-step explanation:
5(4)(-5)+(-2)(4)(-5)+(-5)^2
-100+40+25
-35