Answers
Answer and step-by-step explanation:
The polar form of a complex number \(a+ib\) is the number \(re^{i\theta}\) where \(r = \sqrt{a^2+b^2}\) is called the modulus and \(\theta = tan^-^1 (\frac ba)\) is called the argument. You can switch back and forth between the two forms by either remembering the definitions or by graphing the number on Gauss plane. The advantage of using polar form is that when you multiply, divide or raise complex numbers in polar form you just multiply modules and add arguments.
(a) let's first calculate moduli and arguments
\(r_1 = \sqrt{(-2\sqrt3)^2+2^2}=\sqrt{12+4} = 4\\ \theta_1 = tan^-^1(\frac{2}{-2\sqrt3}) =-\pi/6\\r_2=\sqrt{1^2+1^2}=\sqrt2\\ \theta_2 = tan^-^1(\frac 11)= \pi/4\)
now we can write the two numbers as
\(z_1=4e^{-i\frac \pi6}; z_2=e^{i\frac\pi4}\)
(b) As noted above, the argument of the product is the sum of the arguments of the two numbers:
\(Arg(z_1\cdot z_2) = Arg(z_1)+Arg(z_2) = -\frac \pi6 + \frac \pi4 = \frac\pi{12}\)
(c) Similarly, when raising a complex number to any power, you raise the modulus to that power, and then multiply the argument for that value.
\((z_1)^1^2=[4e^{-i\frac \pi6}]^1^2=4^1^2\cdot (e^{-i\frac \pi6})^1^2=2^2^4\cdot e^{-i(12)\frac\pi6}\\=2^2^4 e^{-i\cdot2\pi}=2^2^4\)
Now, in the last step I've used the fact that \(e^{i(2k\pi+x)} = e^i^x ; k\in \mathbb Z\), or in other words, the complex exponential is periodic with \(2\pi\) as a period, same as sine and cosine. You can further compute that power of two with the help of a calculator, it is around 16 million, or leave it as is.
Related Questions
solve the system find the solution
Answers
y = 6x - 1 ------------------------(1)
y = 3x - 4 ----------------------(2)
substitute (1) in equation(2)
6x - 1 = 3x - 4
collect like term
6x - 3x = -4 + 1
3x = - 3
Divide both-side of the equation by 3
x = -1
substitute x = -1 into equation (2) and then solve for y
y = 3(-1) - 4
y = - 3 - 4
= -7
Hence
Ronnie made a scale drawing of a shopping center. A bakery in the shopping center is 7 inches wide in the drawing. The actual bakery is 42 feet wide. What is the scale of the drawing?
Answers
Answer:
1:72
Step-by-step explanation:
The scale of a drawing is the ratio of the length in the drawing to the real length.
Drawing length: 7 inches
Real length: 42 ft × 12 in. / ft = 504 in.
Scale:
7 in. to 504 in.
Divide both numbers by 7:
1 in. to 72 in.
Scale: 1:72
Barry Katz earns $1,275 per week. He is married and claims two exemptions. What is Barry’s income tax? Use percentage method.
Answers
Answer: I’m not sure but I think the answer is $255
Step-by-step explanation:
For each linear operator T on the vector space V, find an ordered basis for the T-cyclic subspace generated by the vector z. (a) V=R^4, T(a,b,c,d) = (a I b,b - c,a + c,a + d), and z = e1.(b) V=P3(R), T(F(x)) = f'(x), and z = x^3 (c) V=M2x2(R), T(A) = A^t, and z = (0 1 1 0) (d) V=M2x2(R), T(A) =(1 2 2 2) A, and z =( 0 1 1 0)
Answers
The ordered bases for the T-cyclic subspaces generated by z in (a), (b), (c), and (d) are\(B={e1, Te1, T^2e1, T^3e1}, B={x^3, 3x^2, 6x, 6}, B={(0 1 1 0), (1 0 0 1), (0 1 -1 0), (-1 0 0 -1)}\) and \(B={(0 1 1 0), (1 2 2 2), (2 3 3 3), (3 4 4 4)}\), respectively.
a)The T-cyclic subspace generated by z = e1 = (1,0,0,0) is spanned by the vectors e1, \(Te1=(1,b,-c,a+d), T^2e1=(b,-c,a+c,2a+d)\) and \(T^3e1=(-c,a+c,2a+d,3a+2d)\). So the ordered basis for the T-cyclic subspace generated by e1 is \(B={e1, Te1, T^2e1, T^3e1}\).
b)The T-cyclic subspace generated by \(z=x^3\) is spanned by the vectors \(x^3, Tx^3=3x^2, T^2x^3=6x, T^3x^3=6\). So the ordered basis for the T-cyclic subspace generated by x^3 is \(B={x^3, 3x^2, 6x, 6}\).
c)The T-cyclic subspace generated by z=(0 1 1 0) is spanned by the vectors (0 1 1 0), \(T(0 1 1 0)=(1 0 0 1), T^2(0 1 1 0)=(0 1 -1 0), T^3(0 1 1 0)=(-1 0 0 -1)\). So the ordered basis for the T-cyclic subspace generated by (0 1 1 0) is \(B={(0 1 1 0), (1 0 0 1), (0 1 -1 0), (-1 0 0 -1)}\).
d)The T-cyclic subspace generated by z=(0 1 1 0) is spanned by the vectors (0 1 1 0). So the ordered basis for the T-cyclic subspace generated by (0 1 1 0) is \(B={(0 1 1 0), (1 2 2 2), (2 3 3 3), (3 4 4 4)}\).
The ordered bases for the T-cyclic subspaces generated by z in (a), (b), (c), and (d) are \(B={e1, Te1, T^2e1, T^3e1}, B={x^3, 3x^2, 6x, 6}, B={(0 1 1 0), (1 0 0 1), (0 1 -1 0), (-1 0 0 -1)}\), and \(B={(0 1 1 0), (1 2 2 2), (2 3 3 3), (3 4 4 4)}\), respectively.
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A 12 _ foot ladder is leaning against a house the ladder reaches 7.7 feet up the wall wha is the angle of elevation of the ladder?
Answers
Check the picture below.
Make sure your calculator is in Degree mode.
please help me i’ll give you brainlist!
Answers
X intercept: 10/5
Y intercept: 0
How to find intercept of a linear equation?To find the intercepts of a linear equation, you need to set either the X or Y variable to 0 and then solve for the other variable.
For example, if you have the equation 5x - 2y = 10, you can set X equal to 0 and solve for Y to find the Y intercept, or you can set Y equal to 0 and solve for X to find the X intercept.
X Intercept:
To find the X intercept, set Y equal to 0 and solve for X.
5x - 2(0) = 10
5x = 10
x = 10/5
X intercept = 10/5
Y Intercept:
To find the Y intercept, set X equal to 0 and solve for Y.
5(0) - 2y = 10
-2y = 10
y = -10/2
Y intercept = 0
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Given the system of equations. What is the X-value in the solution?
8x+ 3y = 7
4x+ 3y = 5
Answer
Answers
Answer:
x= 1/2
Step-by-step explanation: 8x+3y=7
-(4x+3y=5)
4x=2, and you get x=1/2
Sherelle and Venita find two bags of coins in their grandmother’s attic. Each girl selects a bag, takes a random sample of 20 coins from that bag, and lists the dates of those coins as the number of years since 1900. Here are the results.
Sherelle: 72, 56, 81, 84, 75, 26, 39, 62, 65, 66, 85, 71, 74, 68, 73, 66, 83, 60, 58, 72
Venita: 45, 65, 73, 57, 79, 66, 69, 75, 51, 44, 53, 69, 58, 77, 51, 55, 53, 78, 70, 62
1.Find the five-number summary of each data set. Then construct box-and-whisker plots of the data sets. Draw both plots above the same number line. Label the plots with the girls’ names.
Answers
Answer:
Step-by-step explanation:
Hello!
The five-number summary comprehends the minimum and maximum values, the 1st and 3rd quartiles and the median.
1st step: is to arrange the data sets from least to greatest.
Sherelle:
26 39 56 58 60 62 65 66 66 68 71 72 72 73 74 75 81 83 84 85
Venita:
44 45 51 51 53 53 55 57 58 62 65 66 69 69 70 73 75 77 78 79
2nd step: To calculate each quartile you have to determine their position and then identify which observation sits in that position:
Quartile 1:
PosQ₁: n/4= 20/4= 5
Quartile 2 (Median)
PosMe: n/2= 20/2= 10
Quartile 3
PosQ₃: n*(3/4)= 20*(3/4)= 15
Since both samples have the same size, the 1st quartile will be the fifth observation of each sample, the median will be the tenth observation and the 3rd quartile will be the fifteenth.
Sherelle:
Q₁: 60
Me: 68
Q₃: 74
Minimum: 26
Maximum: 85
Venita:
Q₁: 53
Me: 62
Q₃: 70
Minimum: 44
Maximum: 79
I hope this helps!
Answer:
Step-by-step explanation:
4 x (50 – 1)
4 x 50
50 - 4 x 1
-
Answers
Answer: 4 times (50-1)=196
4 times 50 =200
50-4 times 1 =46
Step-by-step explanation:
First, determine if the three side lengths could form a triangle. (Recall from earlier, the sum of the two smaller sides must be greater than the third side). If yes, classify the triangle further as acute, right, or obtuse.17 ,1712 , Not a triangle
acute
right
obtuse
Answers
The triangle of given dimensions is isosceles and acute triangle.
Triangle is a shape with three sides and hence three angles. We know that triangles are classified according to the sides and angles. Based on angles, the classification of triangle is acute, obtuse and right triangle.
Based on sides, the classification of triangle is equilateral, isosceles and scalene triangle. Now, if the dimensions will form a triangle or not can be estimated through the formula -
side 1 + side 2> side 3
We see that 17 + 17 > 12
34 > 12
Thus, the triangle will form.
Moreover the two sides are equal hence it will be isosceles triangle. Since two sides are equal thus two angles will also be equal.
Now, if we square the sides and find their sum, it will be greater than square of third side. Hence, it is acute triangle.
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Using the table of integrals, solve
Answers
The table of integrals can be a useful tool for simplifying the integration process, but it's important to have a solid understanding of calculus concepts in order to use it effectively.
The table of integrals is a tool used in calculus to simplify the process of finding antiderivatives. It lists various functions and their corresponding antiderivatives, or integrals.
To use the table of integrals, you first need to identify the function you are trying to integrate. Then, find a similar function in the table and use the corresponding antiderivative to solve your integral.
It's important to note that not all functions have a corresponding antiderivative listed in the table, and in those cases, other integration techniques must be used.
Additionally, it's always a good idea to double check your work and make sure the answer makes sense in the context of the original problem.
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If we use the table of integrals, the solution would be In | tan(x+7) + sec(x+7)|+C. Option A
How do we solve function using table of integral?Using the table of integral, we say
∫ (1/cos(x +7)) dx ⇒ ∫sec (x+7) dx
u = x+7 ⇒ du = dx
= ∫sec (u) du
Expand the fraction by tan (u) + sec (u)
= ∫(sec(u) (tan(u) + sec (u))/ tan (u) + sec (u)
= ∫((sec(u) tan(u) + sec²(u))/ (tan (u) + sec (u))) du
Substitute v= tan(u) + sec (u) ⇒ dv = (sec(u) tan(u) + sec²(u))du
= ∫(i/v)dv
= In(v)
Undo substitution v = tan(u) + sec(u)
=In(tan(u) + sec(u))
= In(tan(x+7) + sec(x +7))
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For the given scenario, determine the type of error that was made, if any.
A new whitening toothpaste advertises four shades as the mean number of shades the toothpaste whitens your teeth. One user claims that the mean number of shades the toothpaste whitens your teeth is less than four shades. The user conducts a hypothesis test and fails to reject the null hypothesis. Assume that in reality, the mean number of shades the toothpaste whitens your teeth is three shades. Was an error made? If so, what type?
a. yes type I error
b. yes type II error
c. no correct decison
Answers
Answer:
b. yes type II error
Step-by-step explanation:
When the null hypothesis is false as is in this case and it is accepted so the decision made is wrong. This is a type II error.
When we accept the null hypothesis when actually it is false it is called a type II error.
In reality, the mean number of shades the toothpaste that whitens the teeth is three shades.
Where as the null hypothesis was that the four shades are taken as the mean number of shades the toothpaste whitens the teeth.
Therefore as the null hypothesis which is actually false is accepted so a wrong decision is made a type II error is made.
What is the circumference of a circle with a radius of 14 feet use 22/7 for pi
2,464?” Feet
616 feet
88 feet
44 feet
Answers
The circumference of a circle with a radius of 14 feet is 88 feet.
What precisely is a circle?Every point in the surface of a circle, which is a circular, two-dimensional object, is evenly separated from the centre. All lines that cross the circle join together to produce the line of reflected symmetry. Each angle has axis of symmetry around the centre, in addition. Monitoring a linear interpolation in a surface while keeping a fixed distance from some other point will result in the closed shape of a circle. The English term circle derives from the Greek word curriculum provides students, which also means hoop or ring.
When we find the circumference, we obtain:
The circumference( C) of a circle is 2πr.
C=2πr
⇒C=2×\(\frac{22}{7}\)×14=88 feet.
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square root of 5, 0.6, 4/9, 0.7, square root of 2
Answers
2: 0.77
3: 2/3 or 0.6
4: 0.83
5: 1.41
A box is 20 cm long. Which of these is closest to the length of this box in feet?
Answers
Answer: 0.66 Feet
Step-by-step explanation: There are 2.54 centimeters in an inch and there are 12 inches in a foot.
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What is the equation of the line that passes through the points given in the table
Answers
Answer:
B: y = -2x + 5
Step-by-step explanation:
-2 - 1 = -3
9 - 3 = 6
\(\frac{6}{-3}\) = -2
gradient = -2
3 = 1 x -2 + c
3 = -2 + c
5 = c
y intercept = 5
equation = y = -2x + 5
so the answer is option b
Which value of a from the set (16,18,20,22) makes the equation 2(42-a)=48 true
Answers
Steps to solve:
2(42 - a) = 48
~Distribute left side
84 - 2a = 48
~Subtract 84 to both sides
-2a = -36
~Divide -2 to both sides
a = 18
Best of Luck!
The options are Point E
Point D
Point F
Point C
Answers
Answer:
F
E
D
C
Step-by-step explanation:
The point that is not a solution to any of the equations is the point that no lines pass through. Therefore, this point is Point F.
The point that is a solution to only the equation x+y=4 is the point that only line B passes through. Therefore, this point is Point E.
The point that is a solution to only the equation 5x+10y=25 is the point that only line A passes through. Therefore, this point is Point D.
The point that represents the solution to the system is the point that both lines pass through. Therefore, the point is Point C.
Determine the polynomial y(x) that has roots at x=-0.7, x=0.5, x=3.4 and x=5.8
Make a plot of the polynomial in the domain -1<=x<=6
Answers
The polynomial with given roots is,
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902.
What is factor of polynomial?
In mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.
Given:
The polynomial y(x) that has roots at x = -0.7, x = 0.5, x = 3.4 and x = 5.8.
Since if 'a' is a root of the polynomial then (x - a) is the factor of the polynomial.
So, the factors of the polynomial are, (x + 0.7), (x - 0.5), (x - 3.4), (x - 5.8).
⇒ y(x) = (x + 0.7)(x - 0.5)(x - 3.4)(x - 5.8)
= (x^2 - 0.5x + 0.7x - 0.35)(x^2 - 5.8x - 3.4x + 19.72)
= (x^2 + 0.2x - 0.35)(x^2 - 9.2x + 19.72)
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902
The graph of the polynomial on the domain -1 ≤ x ≤ 6 is attached below.
Hence, the polynomial with given roots is,
y(x) = x^4 - 9x^3 + 17.53x^2 + 7.164x - 6.902.
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Directions - Write an equation in slope-intercept (y = mx + b) form for each line.
Answers
The equation in slope intercept form of the line that passes through the graph is y = 2/3x + 2
How to determine the equation of the line?From the graph, we have the points to be given as
(-3,0) and (0,2)
Start by calculating the slope of the points using the following slope equation
m = (y₂ - y₁)/(x₂ - x₁)
Where
(x, y) = (-3,0) and (0,2)
So, we have
m = (2 - 0)/(0 + 3)
Evaluate
m = 2/3
The equation is then calculated as
y = m(x - x₁) + y₁
Where
(x, y) = (0, 2) and m = 2/3
So, we have
y = 2/3(x - 0) + 2
Expand
y = 2/3x + 2
Hence, the equation in slope intercept form of the line is y = 2/3x + 2
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Need this answered as soon as possible please!! Flo is designing a rectangular billboard using a scale drawing. the billboard is 6 m long. Flo draws a rectangle 12 cm long. The area of Flo's rectangle is 72 cm squared. What is the area of the billboard in square meters?
Answers
sale of eggs that are contaminated with salmonella can cause food poisoning in consumers. a large egg producer takes an srs of 200 eggs from all the eggs shipped in one day. the laboratory reports that 9 of these eggs had salmonella contamination. unknown to the producer, 0.1% (one-tenth of 1%) of all eggs shipped had salmonella. identify the population, the parameter, the sample, and the statistic.
Answers
The simple random sample of 200 eggs
Statistic=Proportion of eggs in the sample that contain salmonella.
p = 0.045
The population is the individuals that are being studied.
population=All eggs in one day
A sample is the part of the population of which information was actually collected.
Sample=The simple random sample of 200 eggs
A parameter is a descriptive measure for a population.
Parameter=Proportion of eggs shipped that day that contain salmonella.
p = 0.1 % =0.001
A statistic is a descriptive measure for a sample.
STATISTIC=Proportion of eggs in the sample that contain salmonella.
p=9÷200 = 0.045
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A=(a+b)h divided by2
Answers
In your own words, explain the difference of independent and dependent events.
Answers
Answer:
Independent means not having to depend on other people, i am myself and dont need no body and being dependent I will need someone by my side because i can't do it alone without some one by my side.
Step-by-step explanation:
Which descriptions offer examples of Security and Protective Services workers? Select all that apply O Angela analyzes information to prevent gang activity and drug trafficking O Rosamund writes tickets and warnings to vehicles that violate parking regulations Andre inspects forests for hazards and identifies ways to prevent fires O Kalil manages security workers and policies for a company O Leilani works in an airport, inspecting people and baggage to make sure they meet safety regulations, O Jeremiah records official court sessions, including all that is said and done.
Answers
Answer:
the one above is definitive wrong ;) ;))))))))
Step-by-step explanation:
i did not get the answer off of him ;)
#edge suks #trash
Laws governing public safety are designed to safeguard employees from potentially fatal situations. Since commercial activity restrictions are the main focus, public safety requirements prioritize the protection of workers.
Thus, the correct option is C.
What do you mean by public safety?Legal professionals refer to people like police officers and firefighters as "public safety officers," with "the protection of the broad public" being the definition of public safety.
Many governments base their policies on the notion of defending the physical welfare of their citizens.
The Act aims to protect the public from the health and safety hazards associated with commercial activity, ensure the health, safety, and welfare of employees and other people at work, and remove workplace dangers at their source.
The Act's goals, which are to "promote and improve public health and wellbeing and to avoid sickness, injury, disability, and early death," are outlined in its aims
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Please, help it is geometry
Answers
Answer:
WR ≈ 13.862
Step-by-step explanation:
using Pythagoras' identity in the right triangle.
the square on the hypotenuse is equal to the sum of the squares on the other 2 sides , that is
WR² + RT² = WT²
WR² + 12.1² = 18.4²
WR² + 146.41 = 338.56 ( subtract 146.41 from both sides )
WR² = 192.15 ( take the square root of both sides )
WR = \(\sqrt{192.15}\) ≈ 13.862 ( to the nearest thousandth )
When Alisa entered a fraction into a calculator, the calculator showed 0.35. What fraction
could Alisa have entered?
Answers
Answer:
7/20
Step-by-step explanation:
The enrollment at a local university increased from 14,000 students to 20,000 students over a six-year period. What was the approximate average percent increase per year in student enrollment at the university?
A. 5%
B. 30%
C. 7%
D. 43%
Answers
The approximate average percent increase per year in student enrollment at the university is = 5%. That is option A.
Calculation of average percent increaseInitial number of student = 14,000 students
Total number of students over 6 years = 20,000 students
The total additional students for 6 years,
= 20,000- 14,000
= 6,000 students.
Therefore every year additional 1000 students where admitted into the local university.
The average percent increase per year in student enrollment at the university,
= 1000/20000 × 100
= 100000/20000
=5%
Therefore, the approximate average percent increase per year in student enrollment at the university is = 5%.
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which of the following is divisible by 9?
Answers
Answer:
If a numbers digits add up till 9
Step-by-step explanation:
Twelve more than five times a number is equal to the difference between 47 and twice the number. Find the number.
Answers
ANSWER
\(5\)EXPLANATION
Let the number be x
Twelve more than 5times the number means;
\(5x+12\ldots\text{.}.\ldots\ldots.1\)Difference between 47 and twice the number means;
\(47-2x\ldots\ldots\ldots\text{.}.2\)Now, equating the two equations we have;
\(5x+12=47-2x\)solving for x, we have;
\(\begin{gathered} 5x+12=47-2x \\ 5x+2x=47-12 \\ 7x=35 \\ \end{gathered}\)
dividing both side by 7, we have;
\(\begin{gathered} \frac{7x}{7}=\frac{35}{7} \\ x=5 \end{gathered}\)Therefore the number is
\(5\)