Answers
let x be the real part of z
let y be the imaginary part of z
[tex]arg(x +i y - 1) = arg(x + iy + 3i) \\ arg(x - 1 + iy) = arg(x + (y + 3)i) \\ arctan( \frac{y}{x - 1} ) = arctan( \frac{y + 3}{x} ) \\ arctan( \frac{ (\frac{y}{x - 1} ) - ( \frac{y + 3}{x}) }{1 +( \frac{y}{x - 1} )( \frac{y + 3}{x} )} ) = 0 \\ arctan( \frac{ \frac{xy - (x - 1)(y + 3)}{x(x - 1)} }{ \frac{x(x - 1) + y(y + 3)}{x(x - 1)} } ) = 0[/tex]
[tex]arctan( \frac{xy - xy - 3x + y + 3}{x {}^{2} - x + y {}^{2} + 3y } ) = 0 \\ \frac{ - 3x + y + 3}{x {}^{2} - x + y {}^{2} + 3y} = tan(0) = 0 \\ - 3x + y + 3 = 0 \\ y = 3x - 3[/tex]
[tex] \frac{x - 1}{y} = \frac{x - 1}{3x - 3} = \frac{x - 1}{3(x - 1)} = \frac{1}{3} \\ [/tex]
Answer:
[tex]\dfrac{x-1}{y}=\dfrac{1}{3}[/tex]
Step-by-step explanation:
Sums of real and imaginary numbers are known as complex numbers:
[tex]\large\boxed{z = x + iy}[/tex]
(where x is the real number and iy is the imaginary number).
Complex numbers can be represented on an Argand diagram.
The x-axis is the real axis.The y-axis is the imaginary axis.z = x + iy is represented on the diagram by the point P(x, y).
[tex]\textsf{Argument of a complex number $= \boxed{\arg z}$}[/tex]
It is the angle between the positive real axis and the line joining that number to the origin on an Argand diagram, measured in an anticlockwise direction.
[tex]\textsf{For \;$z = x + iy$, \;the argument $\theta$ satisfies\; $\tan \theta = \dfrac{y}{x}$.}[/tex]
[tex]\textsf{Given: \quad $\arg (z-1)=\arg(z+3i)$}[/tex]
[tex]\textsf{If $z$ is a complex number then $z = x + iy$}.[/tex]
[tex]\textsf{Substitute this into the given equation}:[/tex]
[tex]\begin{aligned} \arg(z-1)&=\arg(z+3i)\\\implies \arg(x + iy - 1) &= \arg(x + iy + 3i)\\\implies \arg((x - 1) + iy) &= \arg(x + i(y + 3))\end{aligned}[/tex]
[tex]\textsf{If the argument $\theta$ satisfies\; $\tan \theta = \dfrac{y}{x} \implies \theta=\tan^{-1}\left(\dfrac{y}{x}\right)$}:[/tex]
[tex]\implies \tan^{-1}\left(\dfrac{y}{x-1}\right)=\tan^{-1}\left(\dfrac{y+3}{x}\right)[/tex]
[tex]\implies \tan^{-1}\left(\dfrac{y}{x-1}\right)-\tan^{-1}\left(\dfrac{y+3}{x}\right)=0[/tex]
[tex]\boxed{\begin{minipage}{6.2cm}\underline{Trigonometric identity}\\\\ $\tan^{-1}(a)-\tan^{-1}(b)=\tan^{-1}\left(\dfrac{a-b}{1+ab}\right)$\\ \end{minipage}}[/tex]
Use the arctan identity:
[tex]\implies \tan^{-1}\left(\dfrac{\dfrac{y}{x-1}-\dfrac{y+3}{x}}{1+\left(\dfrac{y}{x-1}\right)\left(\dfrac{y+3}{x}\right)} \right)=0[/tex]
[tex]\implies \tan^{-1}\left(\dfrac{\dfrac{xy-(y+3)(x-1)}{x(x-1)}}{1+\dfrac{y(y+3)}{x(x-1)}} \right)=0[/tex]
[tex]\implies \tan^{-1}\left(\dfrac{\dfrac{xy-(y+3)(x-1)}{x(x-1)}}{\dfrac{x(x-1)+y(y+3)}{x(x-1)}} \right)=0[/tex]
[tex]\implies \tan^{-1}\left({\dfrac{xy-(y+3)(x-1)}{x(x-1)+y(y+3)} \right)=0[/tex]
[tex]\implies \tan^{-1}\left({\dfrac{xy-xy+y-3x+3}{x^2-x+y^2+3y} \right)=0[/tex]
[tex]\implies \tan^{-1}\left({\dfrac{y-3x+3}{x^2-x+y^2+3y} \right)=0[/tex]
[tex]\implies \dfrac{y-3x+3}{x^2-x+y^2+3y}=\tan(0)[/tex]
[tex]\implies \dfrac{y-3x+3}{x^2-x+y^2+3y}=0[/tex]
[tex]\implies y-3x+3=0[/tex]
[tex]\implies y=3x-3[/tex]
[tex]\textsf{Substitute\;\; $y=3x-3$ \;\;into \;\;$\dfrac{x-1}{y}$}:[/tex]
[tex]\begin{aligned}\implies \dfrac{x-1}{y}& =\dfrac{x-1}{3x-3}\\\\ & =\dfrac{x-1}{3(x-1)}\\\\ & =\dfrac{1}{3}\end{aligned}[/tex]
[tex]\textsf{Therefore, the value of\; $\dfrac{x-1}{y}$\; is\; $\dfrac{1}{3}$}.[/tex]
Related Questions
Can someone please help me I will mark u brilliant
Answers
(Blank space filler)
Answer:
M would be at (0,-2)
Step-by-step explanation:
The question asks you to rotate the image 180 degrees clockwise. Clockwise means towards the right, such as how a clock moves. Each axis is 90 degrees, so by moving it twice would land at (0,-2).
A horizontal line (such as the x-axis) is 180 degrees, so by mirroring the point, you would see the point is at (0,-2).
What is the correct graph for:
equation: y= x^2 - 1
Answers
Answer:
The first graph.
Hope this helps :)
Step-by-step explanation:
The y-intercept in the equation y = x^2 - 1 is -1.
So the first graph represents the point where the line intercepts the y-axis which is at - 1.
The y-intercept for the second graph is not - 1.
So the first graph would be correct for the equation
So the first graph would be correct for the equation y = x^2 - 1
What geométric symbol would be represented by a baseball bat
Answers
Answer: i think the overall shape of a baseball bat would be more rectangular
Step-by-step explanation:
reason for thinking so: because a baseball bat depending on size, would normally fit into a rectangular box
Plot the intercepts to graph the equation.
2x−8y=−8
Graph it
Answers
Answer:
Graphed it for you!
Step-by-step explanation:
In the figure below, parallelogram ABCD was
translated to parallelogram A'B'C'D'.
How many units and in which direction were
the x-coordinates of parallelogram ABCD
moved?
A. Left 3
B. Right 3
C. Left 7
D. Right 7
Answers
Answer:
C. Left 7
Step-by-step explanation:
The question asks us about the number of units the x-coordinates of the parallelogram moved, therefore we have to look for the horizontal translation of the points A, B, C, and D.
To do this, we can look at the position of any of these points and the position of its corresponding point, to figure out how many units to the left or right they have moved.
For example, let's look at A and A':
To go from A to A', we have to move 7 units to the left.
It's the same for B and B' as well as the other points and their corresponding points.
Therefore the x-coordinates of parallelogram ABCD have moved 7 units to the left.
If Sarah starts with $10 and spends all but $3, how much money does she have left?
Complete the following subtraction exercises.
10 – 2 =
14 – 6 =
15 – 9 =
17 – 8 =
13 – 5 =
11 – 8 =
20 – 8 =
16 – 7 =
12 – 9 =
21 – 9 =
11 – 6 =
5 – 5 =
4 – 0 =
16 – 8 =
10 – 5 =
18 – 7 =
13 – 8 =
12 – 4 =
Solve each of the following subtraction problems.
a. 6,321 – 4,879 =
b. 451 – 275 =
c. 15,278 – 6,327 =
d. 23,000 – 14,995 =
e. 54 – 19 =
On February 1, the electricity meter reading for the Smith residence was 19,423 kilowatt hours. On March 1, the meter read 20,288 kilowatt hours. On April 1, the meter read 21,163 kilowatt hours.
a. How many kilowatt hours of electricity did the Smiths use during February?
b. How many kilowatt hours did they use during March?
Last year, Terry earned $19,436. This year, Terry got a raise that increased her income to $23,457.
a. How much more money did Terry make this year?
b. If Terry gets the same raise again next year, what will her total income be then?
The height of the Leibnitz Range on the moon is 35,000 feet. The earth’s highest point, Mount Everest, measures 29,028 feet. How much taller is the Leibnitz Range?
On a word processing test, Iko typed 694 words in 15 minutes. On that same test six days later, Iko typed 753 words in 15 minutes. By how many words did Iko’s speed improve from the first test to the second test?
Perform each operation. Check by using the inverse operation.
a. 769 – 362 =
b. 502 + 17 =
c. 7,802 – 3,461 =
d. 1,432 + 2,568 =
Answers
Hi!
#1: Sarah would have $3 left.
#2:
10-2=8
14-6=8
15-9=6
17-6=11
13-5=8
11-8=3
20-8=12
16-7=9
12-9=3
12 – 9 = 3
21 – 9 = 12
11 – 6 = 5
5 – 5 = 0
4 – 0 = 4
16 – 8 = 8
10 – 5 = 5
18 – 7 = 11
13 – 8 = 5
12 – 4 = 8
#3:
a. 6,321 – 4,879 = 1,442
b. 451 – 275 = 176
c. 15,278 – 6,327 = 8,951
d. 23,000 – 14,995 = 8,005
e. 54 – 19 = 35
#4
a. How many kilowatt hours of electricity did the Smiths use during February?
They used 19,423 (I believe)
b. How many kilowatt hours did they use during March?
They Used 865
#5:
Last year, Terry earned $19,436. This year, Terry got a raise that increased her income to $23,457.
a. How much more money did Terry make this year?
39,893 (I believe)
b. If Terry gets the same raise again next year, what will her total income be then?
63,350 ( i believe)
#6: 5,972 Ft taller
#7: 59 words.
#8:
a. 769 – 362 = 407
b. 502 + 17 = 519
c. 7,802 – 3,461 = 4,341
d. 1,432 + 2,568 = 4,000
Simplify each expression. (9-7)^3 * 5^0 + (8+3)^2
Answers
Answer:
If your looking for the value it’s 129
(9-7)^3=8
5^0 + (8+3)^2=122
Step-by-step explanation:
Triangle p has b en rotated onto triangle Q
How many degrees anticlockwise has triangle P been rotated?
How many degrees clockwise has triangle P been rotated?
Answers
a. triangle P been rotated by 90 degree in anticlockwise direction
b. triangle P been rotated by 270 degree in clockwise direction
What is angle of rotation ?
In mathematics, the angle of rotation is a measurement of the amount, of namely angle, that a figure is rotated about a fixed point, often the center of a circle.
a. if we rotate the triangle P at 90 degree, from point O at anticlockwise direction, we get triangle Q
b. if we rotate triangle P at 270 degree from point O at clockwise direction, we get triangle Q
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Please Answer Math 10 Part 2
Answers
Answer:
Hello,
Step-by-step explanation:
4)
[tex]u_n=1+3*n\\u_{12}=1+3*12=37\\[/tex]
5)
[tex]u_n=-14+4*n=\\u_{10}=-14+4*10=26\\[/tex]
6)
[tex]u_n=\dfrac{5}{3} +\dfrac{n}{3} \\\\u_{16}=\dfrac{5}{3} +\dfrac{16}{3} =\dfrac{21}{3} =7[/tex]
4. 37
5. 26
6. 7
Refer to the diagram to the right.
a. Write an equation for the diagram to the right.
b. Find the sum.
C.
d.
Describe the sum in terms of the distance from the first addend. Explain.
What integers do the arrows represent?
0
TTTTTTTT
Answers
Answer:
Where is the diagram?
Step-by-step explanation:
Evaluate the expression.
-2/3(1.5÷0.25÷5/4)-5/6
Answers
The value of the given expression is -121/ 30.
PEMDAS stands for parentheses, exponents, multiplication, division, addition, then subtraction. It is basically the rule of the order in which a mathematical expression must be solved.
Here, we are given an expression- -2/3 (1.5 ÷ 0.25 ÷ 5/4) -5/6
It can be written as -2/3 (15/10 ÷ 25/ 100 ÷ 5/4) -5/6
= -2/3 (3/2 ÷ 1/4 ÷ 5/4) -5/6
As per the rules of PEMDAS, we first deal with the terms inside the bracket-
(3/2 ÷ 1/4 ÷ 5/4) = (3/2 × 4 × 4/5)
= (3 × 2 × 4/5)
= 24/5
Thus, the expression becomes-
-2/3 (24/5) -5/6
= -48/ 15 - 5/6
LCM of 15 and 6 is 60
⇒ -(48 × 4)/ 60 - (5 × 10)/ 60
-192/ 60 - 50/ 60
= (-192 - 50)/ 60
= -242/ 60
= -121/ 30
Thus, the value of the given expression is -121/ 30.
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Home Depot entered fiscal 2017 with a total capitalization of $21,904 million. In 2017, debt investors received interest income of $877 million. Net income to shareholders was $8,654 million. (Assume a tax rate of 21%.)
Calculate the economic value added assuming its cost of capital is 10%. (Do not round intermediate calculations. Enter your answer in millions rounded to 2 decimal places.)
Answers
The economic value added (EVA) for Home Depot in the fiscal year, 2017, is $6,463.60 million.
What is the economic value added?Economic value added (EVA) measures a company's financial performance or economic profit.
EVA is based on the residual wealth of the company for the fiscal year. It measures the value a company generates from its invested funds at the cost of capital.
EVA is calculated by deducting the cost of capital from the operating profit after tax (NOPAT).
Data and Calculations:2017 total capitalization = $21,904 million
2017 interest paid to debt investors = $877 million
Net income to shareholders = $8,654 million
Corporate tax rate = 21%
Profit before taxes = $10,954 million ($8,654/ 1 - 0.21)
Cost of capital = 10%
= $2,190.4 million ($21,904 x 10%)
EVA = NOPAT - (Invested Capital * WACC)
= $8,654 million - $2,190.4 million
= $6,463.60 million.
Thus, the economic value added (EVA) for Home Depot in the fiscal year, 2017, is $6,463.60 million.
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b. A lap pool has a length of 27 yards, a width of 23 yards, and a depth of 5 yards. Find the pool's surface area (the water surface) and the total volume of water that the pool holds.
Answers
The volume is 3105 and the surface area is 1742
What is the volume of and what is the surface area?The given parameters are
Length = 27 yards
Width = 23 yards
Height = 5 yards
The volume is
Volume = Length * width * height
So, we have
Volume = 27 * 23 * 5
Evaluate
Volume = 3105
The surface area is
A =2 *(LW + LH + WHW)
So, we have
A = 2 * (27 * 23 + 27 * 5 + 23 * 5)
Evaluate
A = 1742
Hence, the volume is 3105 and the surface area is 1742
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Hey beinggreat78, the ones with a large amount of points are for u and they also mention your name so its easier to spot them (not that i dont want u to answer all the questions)
Answers
Answer:
C.
Step-by-step explanation:
[tex]3 \div 10 = 0.3 < 0.4[/tex]
Answer:
C
Step-by-step explanation:
All work and the answer is provided in the attached screenshot! :)
If w = 18, what is the value of 3w − 11?
A: 307
B: 65
C:43
D: 10
Answers
Answer: 43 ==> C
Step-by-step explanation:
3w − 11=
3(18)-11=
54-11=43 ==> C
Answer:
c, 43
Step-by-step explanation:
3 times 18, minus 11
A friend comes to visit and mentions the temperature in their hometown is currently 25 degrees
Celsius. What is this in Fahrenheit? (Write your answer as a whole number.)
Answers
Answer:
77
Step-by-step explanation:
explanation:
The formula to convert Celsius to Fahrenheit is given by °F = °C × (9/5) + 32
F = [ C × (9/5) + 32 ]
Given that, C = 25
F = 25 × (9/5) + 32
F = 45 + 32
F = 77
3.5X divided by 9 = 4.2 divided by 5. What is X.
I came up with 54.3
Answers
Answer:
2.16
Step-by-step explanation:
if I understand the description correctly:
3.5*x / 9 = 4.2 / 5
3.5x = 4.2*9 / 5
x = 4.2*9 / 5*3.5
x=2.16
what is the answer to 0.25 x 2.6 this is for hw
Answers
please refer to the image
Answers
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
1
2
3
4
5
6
-1
-2
-3
-4
-5
-6
u
p(u)
Write an equation for the function graphed above.
Answers
PLEASE HELP URGENT
Find the value of the variables. Then find the measure of each angle in the diagram.
Answers
The value in fig 4 x = 18.7 and γ = 12.72 . The value in fig 5 x = 10 and γ = 40 .
In figure 4 .
Solving linear equation in two variable
10 γ + 4 x - 22 = 180
3 γ + 11 + 7 x + 4 = 180
10 γ + 4 x = 202 ... eq 1
3 γ + 7 x = 169 ... eq 2
mult. ... eq 1 by - 3 and ... eq 2 by 10 .
- 30 γ - 12 x = - 606
30 γ + 70 x = 1690
Adding above 2 eqns
58 x = 1084
x = 18.7
γ = 12.72
Measurement of angles : 127.2 degrees , 52.8 degrees 134.9 degrees , 49.16 degrees .
In fig . 5
4 x + 13 x + 10 = 180
17 x = 170
x = 10
2 ( γ + 25 ) + 2 γ - 30 = 180
4 γ + 20 = 180
4 γ = 160
γ = 40
Measurement of angles are 130 degrees , 50 degrees , 130 degrees , 50 degrees .
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20 chocolates in a box, 2 are coconut cream, what is the probability that after eating two chocolates in a row that they are not coconut cream
Answers
Answer:
18/20 * 17/19 ≈ 0.805263158
Step-by-step explanation:
18/20 for the first chocolate not to be coco cream
then, the number of non-coco ones is lowered by 1 and the number of total too, so
17/19 for the second chocolate not to be coco cream
The chance that neither of the 2 chocolates is coconut cream is 18/20*17/19 ≈ 0.805263158
I hope this helps! Have an awesome day! : D
36x+9 equals what
big blak vjndjgd
Answers
We can conclude that the equation 36x + 9 equals -1/4.
What exactly is an equation?An equation, in its most basic form, is a mathematical statement demonstrating that two mathematical expressions are equal.3x + 5 = 14, for example, is an equation in which the expressions 3x + 5 and 14 are separated by a 'equal' sign.So,
36x + 936x + 9 = 036x = -9x = -9/36x = -1/4Therefore, we can conclude that the equation 36x + 9 equals -1/4.
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(-5.2)(-7.8) what is it
Answers
Answer:
40.56
Step-by-step explanation:
:)
There are 50 minutes left before the event begins. You have 20 balloons to inflate before the event. How much time can you spend inflating each one equally?
Answers
Answer: 24 seconds
Step-by-step explanation:
20/50=0.4
0.4*60=24
What’s the domain and range of y = |x4|
Answers
Domain: (-∞,∞) {x,x∈R }
Range: [0,∞)
Solution :
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Interval Notation
Interval notation is a method to represent an interval on a number line. In other words, it is a way of writing subsets of the real number line. An interval comprises the numbers lying between two specific given numbers. For example, the set of numbers x satisfying 0 ≤ x ≤ 5 is an interval that contains 0, 5, and all numbers between 0 and 5.
Different Types Of Intervals
Intervals can be classified based on the numbers the set comprises. Some sets include the endpoints specified in the notation, while some might partially or not include the endpoints. In general, there are three types of intervals given as,
Open Interval
Closed Interval
Half-Open Interval
Let us understand the interval notation and different types of intervals in detail using solved examples.
Interval Notation:
y = |x4|
(−∞,∞)
Set-Builder Notation:
{x|x∈R}
The range is the set of all valid
y
values. Use the graph to find the range.
Interval Notation:
[0,∞)
Set-Builder Notation:
{y|y≥0}
Determine the domain and range.
Domain:
(−∞,∞),{x|x∈R}
Range:
[0,∞) , {y|y≥0}
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Helps solve Application
Answers
The percentage of 0 - 17 in 2015 is 23.97% and the percentage is decreasing by 0.048% in 2015
The percentage of 0 - 17 in 2015The equation of the function is given as
P(x) = 26x^-0.03
In 2015, x = 15
Substitute x = 15 in the equation P(x) = 26x^-0.03
P(15) = 26 * 15^-0.03
Evaluate the expression
P(15) = 23.97
Hence, the percentage of 0 - 17 in 2015 is 23.97%
How fast is the percentage decreasing in 2015The equation of the function is given as
P(x) = 26x^-0.03
Differentiate the function
P'(x) = -0.03 * 26x^-1.03
In 2015, x = 15
Substitute x = 15 in the equation
P'(15) = -0.03 * 26 * 15^-1.03
Evaluate
P'(15) = -0.048
Hence, the percentage is decreasing by 0.048% in 2015
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Write a rule for the relation shown in the table.
X: 1, 2, 3, 4
Y: 10, 12, 14, 16
Answers
Y starts at 10 and is getting added by 2
what is the domain of h
Answers
Answer:
Hello. The domain is (-5, 5), which translates to option D.
-5 ≤ x ≤ 5
Is 7 hundreds + 4 thousands + 27 ones greater than, less than or equal to 6 thousands + 4 hundreds? *
Answers
Answer:
6 thousands + 4 hundreds
4727>6400
Step-by-step explanation:
4727<6400