there exists a complex number $c$ such that we can get $z 2$ from $z 0$ by rotating around $c$ by $\pi/2$ counter-clockwise. find the sum of the real and imaginary parts of $c$.
Answers
The sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$
The given problem can be solved using algebraic and geometric methods. We can use algebraic methods, such as the equations given in the problem, and we can use geometric methods by visualizing what the problem is asking. To start, let's translate the given problem into mathematical equations. Let $z_0$ be the original complex number. We want to rotate this point by 90 degrees counter-clockwise about some complex number $c$ to get $z_2$. Thus,$$z_2 = c + i(z_0 - c)$$$$=c + iz_0 - ic$$$$= (1-i)c + iz_0.$$We also know that this transformation will rotate the point $z_1 = (z_0 + z_2)/2$ by 45 degrees. Thus, using similar logic,$$z_1 = (1-i/2)c + iz_0/2.$$Now let's use the formula for rotating a point about the origin by $\theta$ degrees (where $\theta$ is measured in radians) to find a relationship between $z_1$ and $z_0$.$$z_1 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c + iz_0/2 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c = (e^{i\theta/2} - 1)z_0/2.$$We can solve for $c$ by dividing both sides by $1-i/2$.$$c = \frac{e^{i\theta/2} - 1}{1-i/2}\cdot\frac{z_0}{2}.$$We can now use the information given in the problem to solve for the sum of the real and imaginary parts of $c$. We know that rotating $z_0$ by 90 degrees counter-clockwise will result in the complex number $z_2$. Visually, this means that $c$ is located at the midpoint between $z_0$ and $z_2$ on the line that is perpendicular to the line segment connecting $z_0$ and $z_2$. We can use this geometric interpretation to solve for $c$. The midpoint of the line segment connecting $z_0$ and $z_2$ is$$\frac{z_0+z_2}{2} = c + i\frac{z_0-c}{2}.$$Solving for $c$, we get$$c = \frac{z_0+z_2}{2} - \frac{i}{2}(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0- (e^{i\theta/2} - 1)(z_0/2)/(1-i/2)).$$We can now find the real and imaginary parts of $c$ and add them together to get the desired answer. Let's first simplify the expression for $c$.$$2c = z_0+z_2 - i(z_0 - (e^{i\theta/2} - 1)\cdot(z_0/2)\cdot(1+i)/2)$$$$= z_0 + z_2 - i(z_0 - z_0(e^{i\theta/2} - 1)(1+i)/4)$$$$= z_0 + z_2 - i(z_0 - z_0e^{i\theta/2}(1+i)/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0(1-e^{i\theta/2})/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0/4(1-e^{i\theta/2} + 1 - i))$$$$= z_0 + z_2 - i(z_0/2(1-\cos(\theta/2) - i\sin(\theta/2)))$$$$= z_0 + z_2 - i(z_0(1-\cos(\theta/2)) + z_0\sin(\theta/2) - i(z_0\cos(\theta/2))/2.$$Now we can find the real and imaginary parts of $2c$ and divide by 2 to get the real and imaginary parts of $c$. We have$$\operatorname{Re}(2c) = \operatorname{Re}(z_0+z_2) - \operatorname{Im}(z_0)(1-\cos(\theta/2)) - \operatorname{Re}(z_0)\sin(\theta/2)$$$$\operatorname{Im}(2c) = \operatorname{Im}(z_0+z_2) - \operatorname{Re}(z_0)(1-\cos(\theta/2)) + \operatorname{Im}(z_0)\sin(\theta/2).$$Thus, the sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$
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Related Questions
Answer for brainliest
Answers
Step-by-step explanation:
The diameter is 6, so the square is also 6 feet by 6 feet. This means the square has an area of 36 feet.
Find the area of the circle:
3.14 * 3^2
3 ^ 2 = 9
3.14 * 9 = 28.26
Now find the area of the edges of the square.
36 - 28.26 = ~7.74 ft
What is x??????????
Answers
Answer:
23.3
Step-by-step explanation:
in an analysis of variance problem if sst = 120 and sstr = 80, then sse is
a. 200
b. 40
c. 80
d. 129
Answers
In an analysis of the variance problem, if sst = 120 and sstr = 80, then sse is given by :d. 129. In an ANOVA problem, the total variation, SSt, is the sum of the variation between groups and the variation within groups
Here is the explanation :
An analysis of variance (ANOVA) is a statistical test used to identify significant differences between the means of two or more groups of data.ANOVA decomposes the total variation in a dataset into two components: the variation between groups and the variation within groups.
ANOVA problem: In an ANOVA problem, the total variation, SSt, is the sum of the variation between groups and the variation within groups.SSt = SSb + SSwwhere SSb is the variation between groups and SSw is the variation within groups.In the given problem:Given, SSt = 120andSSb = 80Then, SSw = SSt - SSb= 120 - 80= 40Therefore, the correct answer is d. 129.
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Pls help me I need it urgently but please give right answer
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Answer:
Hello your handwriting is bad I don't understand
Write 7x2−x+3x3−11
in standard form.
Answers
Answer:
12 - x is the standard form version :)
Cuántos puntos tengo si mis tareas valen 40% son 51 tareas y entregue 31
Answers
sorry, but can you please transate it into English?
I don't understand...
Which value of y makes the equation 55/y = 5 true?
Answers
Answer:
11
Step-by-step explanation:
55/11=5
Giving brainiest explain answer
Answers
Answer:
A.) -5
Step-by-step explanation:
Slope Formula ---> \(m = \frac{y_2 - y_1}{x_2 - x_1}\) m = slope
\(m = \frac{2 - (-3)}{-2 - (-1)}\)
Eliminate negatives:
\(m = \frac{2 + 3}{ -2 + 1}\)
Solve:
\(m = \frac{5}{-1}\)
Simplify:
\(m = -5\)
HELPPP ASAP!!!!
WILL GIVE BRAINLIEST
Find the slope
Answers
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\(slope = \frac{7 - 11}{0 - ( - 1)} \\ \)
\(slope = \frac{ - 4}{1} \\ \)
\(slope = - 4\)
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Answer:
-4
Step-by-step explanation:
-4
PLS HELP! WILL GIVE BRAINLIEST
Answers
Answer: do 2.1 times 100 than that answer times 0.6
Step-by-step explanation:
Chin was shown the graph of a line that contained point (1, 7). He wrote f(x) = 4x + 3 to correctly represent the line. Which of these equations could represent the same line?
y – 7 = 3(x – 1)
y – 1 = 3(x – 7)
y – 7 = 4(x – 1)
y – 1 = 4(x – 7)
Answers
Answer: it’s the third option
Step-by-step explanation:
Answer:
y – 7 = 4(x – 1)
Step-by-step explanation:
Third option
which equation represents a parabola opening to the right with a vertex at the origin and a focus at (9,0)?
Answers
Answer:
y² = 36x
Step-by-step explanation:
Parabola opens right.
Thus equation is from; y² = 4ax
And focus is at (a, 0)
Since we are given focus at (9, 0),it implies that a = 9
Thus, equation of parabola is;
y² = 4 × 9 × x
y² = 36x
6) You go to dinner with some
friends. Your total bill was $56.32.
You have a coupon for 15%. How
much is your total cost if you leave
a 20% tip?
Answers
Answer:
$57.44
Step-by-step explanation:
1. subtract 15% from the bill with coupon:
$56.32 - ($56.32 x 0.15) = $47.87
2. add 20% to this new bill:
$47.87+(47.87 x 0.20) = $57.44
write the equation of the line 3 and passes through the point (1.3)
Answers
The equation of line with slope 3 passes through the point (1,3) is y =3x.
What is straight line?It is a one-dimensional figure created by joining two points. A line is an object in geometry that is indefinitely long and has neither breadth nor depth nor curvature. Since lines can exist in two, three, or higher dimensional environments, they are one-dimensional things. The term "line" can also be used to describe a line segment in daily life that contains two locations that serve as its ends.
How do we write the equation of line?It is a linear algebraic equation with two variables that can be written as
y=mx+c or y-y1 =m(x-x1)
where, c is a constant and m is the slope of line. (x1,y1) are a point lies on the line.
Slope m indicates the stiffness and direction of a line.
given, slope =3, the line passes through (1,3)
the equation will be y-3 = 3(x-1)
y-3 = 3x-3
y=3x
hence, the equation is y=3x
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The number of people etimated to vote in an election wa 7,000. The actual number of people who voted wa 5,600. By what percent did the etimate vary from the actual amount?
75%
1400%
14%
25%
Answers
The estimate of the no. of people to vote varies by 25% from the actual no. of voters turned out.
Finding the per cent variation of estimated votes from the actual turnout
The no. of people estimated to vote in the election = 7000
The actual no. of people who voted = 5600
A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate percent of a number, divide the number by the whole and multiply by 100. Hence, the percentage means, a part per hundred. The word per cent means per 100. It is represented by the symbol “%”.
∴ The per cent by which the estimate vary from the actual turnout is given by,
= [{7000 – 5600}/ {5600}]×100
= [1400/5600]×100
= 0.25×100
= 25%
Therefore, the 25% of the estimate vary from the actual amount.
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what is the coordinates of A and Q
Answers
Answer:
Coordinate of A: -9
Coordinate of Q: 9
Step-by-step explanation:
1 - 10 = -9
1 + 8 = 9
Solve for x.
ax + bx = 10
Answers
ax+Bc=10
x(a+b)=10
Divide both the sides by a+b
X(a+b)/(a+b)=10/a+b
Therefor x=10/a+b
Helen earns $14.75 per hour as a babysitter. She receives time-and-a-half pay for any hours she works over 40 in a week. What are her wages in a week in which sheworked 50 hours?$798.05$811.30$821.79$832.65None of these choices are correct.
Answers
As per given by the question,
There are given that Helen earns per hours rate is $14.75.
That means, the rgular rate is $14.75 per hour.
Now,
To calculate the employees overtime, multiply their regular rate by 1.5.
Then,
She receives time-and-a-halfe pay menas, she receives over time pay.
So,
Helen earns doller with overtime is,
\(14.75\times1.5=22.125\)Now,
When she worked in 50 hours in a week .
Then,
The wages is,
\(50\times22.125=1106.25\)So, Helen received wages whe she worked 50 hours in a week is 1106.25.
Hence, the option fifth is correct.
in a standard deck of cards, what is the probability of drawing a red card or a face card? answer choices are in the form of a percentage, rounded to the nearest whole number. a.) 25% b.) 73% c.) 62% d.) 23%
Answers
The probability of drawing a red card or a face card in a standard deck of cards is 73% or 0.73.
Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that the event will not occur and 1 indicating that the event will definitely occur.
There are 52 cards in a standard deck, and 26 of them are red (half of the cards are red, because there are two colors: red and black).
There are also 12 face cards in a standard deck (3 face cards in each of the 4 suits).
Therefore, the probability of drawing a red card is 26/52 or 0.50 while the probability of drawing a face card is 12/52, or about 0.23.
The probability of drawing a red card or a face card is the sum of the probabilities of drawing each of these events separately
0.50 + 0.23 = 0.73
Hence, the probability of drawing a red card or a face card is approximately 73%.
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what is a diffrence in a linar function and a non line function
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Solving Systems using tables
Mikayla put $212 in a savings account at the beginning of the year. At the end of each month, she withdrew $12 from her account. Mathew put $30 in his savings account at the beginning of the year. At the end of each month, he deposited $14 in his account. Let x represent the number of months which have passed and y represent the amount in each savings account.
A. Write an equation to represent the amount in Mikayla’s savings account.
B. Write an equation to represent the amount in Matthew’s savings account.
C. Below make an input-output tables that show the amount in each account for the first 8 months
D. When will Mikayla and Matthew have the same amount in their accounts?
Mikayla Matthew
X Y X Y
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
Answers
Answer:
i hope this helps
Step-by-step explanation:
3% is the interest rate. Using the formula I=PRT, you can simply derive the rate by putting the given numbers in.
I = 450P = 3,000R = ?T = 5 Years
The equation now goes,450 = 3,000 * R * 5450 = 15,000 * R450/15,000 = R0.03 = R
multiply it by 100,
R = 3%
how do i solve for x on number 16. With the geometric mean
Answers
Answer:
x = 42
Step-by-step explanation:
Brianna bought a compact disk for $10. An instant rebate gave her a discount and she only had to pay $7.50 for the CD. What is the discount, as a percent, that the instant rebate gave her?
Answers
Answer:
25%
Step-by-step explanation:
The discount was $2.50. Dividing $2.50 by $10, we get 0.25, or 25%
PLEASE HELP PELASE
For ΔABC, the measure in degrees of angles A, B, and C are 60, 55, and x + 20 respectively. What is the value of x?
A.
65
B.
45
C.
95
D.
25
Answers
Answer:
B.45
Step-by-step explanation:
A sum of all angles in a triangle is 180°55+60+x+20=18060+55=115
180-115=65
65-20=45
Answer: 45°
\(f(x)=cos(x+1)+sin(x-2)\)
interval: \([-\frac{\pi}{2},\frac{\pi}{2}]\)
subintervals: 12
estimate the area under the curve using left endpoint rectangles.
express your answer in summation notation and then expand the summation into function form. then estimate the summation to the nearest thousandth.
Answers
The final estimate of the area under the curve is 4.192 to the nearest thousandth.
The subinterval length is (π/2 - (-π/2))/12 = π/24. Let x_0 be the left endpoint of the first subinterval, so x_0 = -π/2. The area under the curve in this subinterval can be estimated by the left endpoint rectangle with height f(x_0) = cos(-π/2 + 1) + sin(-π/2 - 2).
The height of the rectangle in the next subinterval can be estimated by the left endpoint with height f(x_0 + π/24) = cos(-π/2 + π/24 + 1) + sin(-π/2 + π/24 - 2), and so on.
In summation notation, the estimate of the area under the curve can be written as:
A = π/24 × [f(x_0) + f(x_0 + π/24) + ... + f(x_0 + 11π/24)]
Expanding the summation, we get the following:
A = π/24 ×× [cos(-π/2 + 1) + sin(-π/2 - 2) + cos(-π/2 + 2π/24 + 1) + sin(-π/2 + π/24 - 2) + ... + cos(-π/2 + 11π/24 + 1) + sin(-π/2 + 11π/24 - 2)]
We can estimate the summation by evaluating the cosine and sine functions for each term in the summation. The final estimate of the area under the curve is:
A ≈ π/24 × [cos(-π/2 + 1) + sin(-π/2 - 2) + cos(-π/2 + 2π/24 + 1) + sin(-π/2 + π/24 - 2) + ... + cos(-π/2 + 11π/24 + 1) + sin(-π/2 + 11π/24 - 2)] ≈ 4.192 to the nearest thousandth.
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Does anyone know this?
Answers
Simplify the expression to standard form
-1/2(8x - 4) + (-5 + 3x)
Answers
Answer:
\( - x - 3\)
Step-by-step explanation:
1) regroup term.
\( - \frac{1}{2} (8x - 4) + 3x - 5\)
2) Simplify 1/2 ( 8x - 4 ) to 8x - 4/2.
\( - \frac{8x - 4}{2} + 3x - 5\)
3) Factor out the common term 4.
\( - \frac{4(2x - 1)}{2} + 3x - 5\)
4) Simplify 4( 2x - 1 ) / 2 to 2( 2x - 1).
\( - 2(2x - 1) + 3x - 5\)
5) Expand by distributing terms.
\( - (4x - 2) + 3x - 5\)
6) Remove parentheses.
\( - 4x + 2 + 3x - 5\)
6) Collect like terms.
\(( - 4x + 3x) + (2 - 5)\)
8) Simplify.
\( - x - 3\)
Therefor, the answer is -x - 3.
I need help identifying if this is a 90,180,270 degree transformation
Answers
let a = a 0 3(a − b) b . use the factorization a = p dp −1 to compute a
Answers
Using the factorization method, we can compute a value of 'a' based on the given equation \(a = a^{0.3}(a - b)^b\). By expressing 'a' as a power of 'p' multiplied by 'dp' raised to the power of -1, we can simplify the equation and calculate the value of 'a' more easily.
Let's rewrite the equation \(a = a^{0.3}(a - b)^b\) as \(a = p(dp)^{-1\). This form allows us to simplify the equation and solve for 'a'. Now, let's expand the equation using the properties of exponents.
\(a = p(dp)^{-1\)
To eliminate the negative exponent, we can rewrite \((dp)^{-1\)as 1/(dp). Thus, the equation becomes:
a = p/(dp)
We can further simplify this expression by canceling out the 'p' terms:
a = 1/d
Now, we have a clear expression for 'a' in terms of 'd'. The value of 'a' depends on the value of 'd', which is not provided in the given equation. To compute a specific value for 'a', we need to know the value of 'd' or have additional information. Without that information, we cannot determine a numerical value for 'a' but can express it in terms of 'd' as a = 1/d.
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Say whether the given function has limit at the point (0,0). If the limit exists, then find it. (a) f(x, y) = 5ry² 3x² + y² (Hint: the parabola z = - y²); (b) f2(x, y) = Vel+V sin(y). (Hint: [√x + √] × [√ √U] ...). [2,3]
Answers
(a) The function f(x, y) does not have a limit at (0,0). (b) No information is provided to determine the limit for f2(x, y).
(a) For the function f(x, y) = 5ry²/(3x² + y²), we can analyze the behavior as (x, y) approaches (0,0). Since the denominator 3x² + y² becomes zero as (x, y) approaches (0,0), we cannot directly evaluate the function at this point. However, by considering the parabola z = -y², we can observe that the function does not approach a specific value and thus does not have a limit at (0,0).
(b) The function f2(x, y) = Vel + Vsin(y) is not well-defined as no information or context is provided for the variables Vel, V, and U. Without this information, it is not possible to determine the limit of the function at (0,0) or any other point.
Therefore, for (a), the function f(x, y) does not have a limit at (0,0), and for (b), no information is given to determine the limit for f2(x, y).
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