A complex number zı has a magnitude z1] = 2 and an angle 6, = 49°.=Express zy in rectangular form, as 21 == a +bi.Round a and b to the nearest thousandth.21 =+iShow Calculator
![A Complex Number Z Has A Magnitude Z1] = 2 And An Angle 6, = 49.=Express Zy In Rectangular Form, As 21](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/iEoIAnOPnio9Ir2ikeWKY7lsF7UnUwrf.jpeg)
Answers
Complex numbers can be written in two forms:
[tex]\begin{gathered} z=a+b\cdot i \\ z=r\cdot e^{i\theta} \end{gathered}[/tex]Where a and b are known as the real and the imaginary part and r and theta are the magnitude and the angle of the number. In this case we are given these last two quantities and we have to find a and b. One way to do this is recalling an important property of the exponential expression above:
[tex]e^{i\theta}=\cos \theta+i\sin \theta[/tex]Then the exponential form of a number is equal to:
[tex]z=r\cdot e^{i\theta}=r\cdot(\cos \theta+i\sin \theta)=r\cos \theta+i\cdot r\sin \theta[/tex]And since we are talking about the same number then this expression must be equal to that given by a and b:
[tex]a+i\cdot b=r\cos \theta+i\cdot r\sin \theta[/tex]Equalizing terms without i and those with i we have two equations:
[tex]\begin{gathered} a=r\cos \theta \\ b=r\sin \theta \end{gathered}[/tex]Now let's use the data from the exercise:
[tex]\begin{gathered} r=\lvert z_1\rvert=2 \\ \theta=\theta_1=49^{\circ} \end{gathered}[/tex]Then we have:
[tex]\begin{gathered} a=2\cdot\cos 49^{\circ} \\ b=2\cdot\sin 49^{\circ} \end{gathered}[/tex]Using a calculator we can find a and b:
[tex]\begin{gathered} a=1.312 \\ b=1.509 \end{gathered}[/tex]Then the answers for the two boxes are 1.312 and 1.509
Related Questions
The graph shows the equation x=y^2 use the slider for a to move the vertical line on the graph. According to the vertical line test, is this equation a function why or why not?
Answers
Explanation
We are given the equation:
[tex]x=y^2[/tex]We are to use the vertical line test to determine if the equation is a function or not
The vertical line test is a graphical method of determining whether a curve in the plane represents the graph of a function by visually examining the number of intersections of the curve with vertical lines.
The typical example below helps give a better explanation
So for the function
[tex]x=y^2[/tex]We can observe that the equation is not a function because the vertical line cuts the graph in more than one point
This is shown below for values of x = and x =8
Rita is applying for a job as an engineer. Her starting salary at Company A will be $80,000 with an $800 yearly raise. Her starting salary at company B will be $65,000 with a 5% increase each year. If Rita is working at a company for 5 years. Which company should she pick?
Answers
Given:
In company A, starting salary is $80,000.
The yearly increment is $800.
So,
80,000+800=80,800
80,800+800=81,600
81,600+800=82,400
82,400+800=83,200
83,200+800=84,000
So, at the 5 year, she will get $84,000
In company B,
The initial salary is $65000 with a 5% increase each year.
So,
[tex]\begin{gathered} 65000\times\frac{105}{100}=68250 \\ 68250\times\frac{105}{100}=71662.5 \\ 71662.5\times\frac{105}{100}=75245.625 \\ 75245.625\times\frac{105}{100}=79007.906 \\ 79007.906\times\times\frac{105}{100}=82958.30 \end{gathered}[/tex]In the 5th year, she will get $82,958.30.
If Rita is working at a company only for 5 years, then she would choose company A. Because she will get salary in company A more than company B.
But, if she works for more than 5 years, she will get a salary in company B more than company A.
16) A bottle of high blood pressure medication contains 90 tablets. Each tablet contains 150 mg of the active ingredient. How many grams of the active ingredient are in the entire bottle? Hint: 1 gram = 1,000 mg
Answers
One tablet contains 150 mg of the active ingredient. Then, 90 tablets contain
[tex]90\text{ tablets }\cdot\frac{150\text{ mg}}{1\text{ tablet}}=13500mg_{}[/tex]1 gram is equivalent to 1,000 mg, then 13500 mg is equivalent to
[tex]13500\text{ mg}\cdot\frac{1\text{ gram}}{1000\text{ mg}}=13.5\text{ grams}[/tex]There are 13.5 grams of the active ingredient in the entire bottle
A right triangle has a hypotenuse of 18 feet and a side length opposite of 12 feet . What is the measure of angle A the nearest degree ?
Answers
can you PLEASE help me
Answers
In this problem we know that
Applying the exterior angle theorem
10x+30=58+(7x-1)
solve for x
10x-7x=57-30
3x=27
x=9Find m
mm
Find mmm
CM Bookmarks Geometry Unit 11 Test Area of Plane Figures E CALCULATOR • colo a a noosa 12. Find the area of the shaded region of the figure
Answers
It is a trapezoid
[tex]\begin{gathered} \text{ Area = h}\frac{B\text{ + b}}{2} \\ \text{Area = 4}\cdot\text{ }\frac{16\text{ + 6}}{2} \\ \text{Area = 4}\cdot\frac{22}{2} \\ \text{Area = }\frac{88}{2} \\ \text{Area = 44 mm}^2 \end{gathered}[/tex]
Acetone (fingernail polish remover ) has a density of 0.7857 g/cm^3.A) what is the mass in grams of 17.56 mL of acetone?B) what is the volume in milliliters of 7.22 g of acetone?
Answers
We can use density as a factor of conversion.
To find the mass in grams of the volume of acetone, multiply the volume by the density (always check the units, that in this case are consistent because 1cm^3=1mL):
[tex]17.56mL\cdot\frac{0.7857g}{mL}=13.79g[/tex]To find the volume of the mass of acetone, divide the mass by the density:
[tex]7.22g\cdot\frac{1mL}{0.7857g}=9.18mL[/tex]The graph of f(x) = 4+ is shown below in blue. This graph in red is a transformation of f(x). Write a function thatdescribes the
Answers
Solution
Step 1:
Write the parent function
[tex]f(x)\text{ = 4}^x[/tex]Step 2:
Transformation of f(x) to g(x)
First, f(x) was reflected across the x-axis
[tex]f(x)\text{ }\rightarrow\text{ -4}^x\text{ }\rightarrow\text{ g\lparen x\rparen}[/tex]Step 3:
Then the function is later shifted 3 units vertically down.
[tex]g(x)\text{ = -4}^x\text{ - 3}[/tex]Final answer
[tex]g(x)\text{ = -4}^x\text{ - 3}[/tex]-20 increased by 4
translating words to algebraic expressions
Answers
The algebraic expression is -20 +4.
What is algebraic expression?
An algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression.
Given, situation -20 increased by 4.
Translate the phrase -20 increased by 4 into an algebraic expression.
You probably already know that more than is associated with addition so the sign is not going to change. But what about the order of the terms?
Think about it this way: we have a number (some unknown value) and this phrase represents -20 increased by whatever that value is. So, in this case, you will start with the number -20 and add 4.
we get -20+4.
To know more about algebraic expression, visit:
https://brainly.com/question/24267361
#SPJ1
If point B, shown on the coordinate plane below, is reflected over the y-axis to create B’, what will be the coordinates of B’?(-5, 2)(5, 2)(-5, -2)(5, -2)
Answers
Solution
- The transformation for reflection over the y-axis is given below:
[tex](x,y)\to(-x,y)[/tex]- We have been given the coordinate of B to be (-5, -2) as shown below:
- Thus, applying the transformation formula given above, we have:
[tex]\begin{gathered} (x,y)\to(-x,y) \\ (-5,-2)\to(-(-5),-2)=(5,-2) \end{gathered}[/tex]- Thus, the reflected point B' is
[tex](5,-2)[/tex]- This is shown below:
9. A coin is tossed and a number cube is rolled. What is the probability of getting tails and rolling a two?
Answers
Okay, here we have this:
Considering the provided information, we are going to calculate the requested probability, so we obtain the following:
Probability of getting tails and rolling a two=Probability of getting tails * Probability of getting a two
And basing ourselves on the fact that when tossing a coin there are two possible events and in this case a favorable one, and when tossing the die there are 6 possible events and one favorable for this case, we have:
Probability of getting tails and rolling a two=1/2*1/6
Probability of getting tails and rolling a two=1/12
Finally we obtain that the probability of getting tails and rolling a two is 1/12.
3. For the sequence defined by tn = 3n + 8,find each indicated term.a) t1b) t7c) t14
Answers
The sequence is given by .
[tex]t_n=3n+8[/tex]a. The indicated term
[tex]t_1=3(1)+8[/tex][tex]t_1=3+8[/tex][tex]t_1=11[/tex]b. The indicated term
[tex]t_7=3(7)+8[/tex][tex]t_7=29[/tex]c. The indicated term
[tex]t_{14}=3(14)+8[/tex][tex]t_{14}=42+8[/tex][tex]t_{14}=50[/tex]
Find the area of this irregular shape.
[Round off to the nearest whole number.]
sq. units
Answers
Answer:
Step-by-step explanation:
number of complete squares=14
number of half or more than half squares=4
whole squares=4/2=2
area≈14+2=16 sq. units
About how far does the flag have to travel to complete full rotation?1. 37.68 ft2. 75.36 ft3. 150.72 ft4. 452.16 ftPlease explain if possible!
Answers
Due to the circular shape, one complete rotation of the flag is equal to the circumference of a circle with radius 12 ft.
Then, use the following formula for the cirumference:
C = 2π·r
where r is the radius. In this case, r = 12 ft and π = 3.14. Replace the previous values into the formula for C:
C = 2(3.14)(12 ft)
C = 75.36 ft
Hence, the flag has to travel 75.36 ft to complete a full rotation
Write an expression in simplest form that represents the income from w women and m men getting a haircut and shampoo. Women: haircut $45 shampoo $12. Men: haircut $15 shampoo $7
Answers
Write an expression in simplest form that represents the income from w women and m men getting a haircut and shampoo. Women: haircut $45 shampoo $12. Men: haircut $15 shampoo $7
the income is equal to
45w+12w+15m+7m
combine like terms
57w+22m
total income=57w+22mwhere
w is the number of women
m is the number of men
the image is downloading nowabout 15%is too slowlyDo you can write the question?Study PathsTestPlacement Test Williston State College 2018 Study PathTestInit: GeometryogressQuestion ID: 1191695The movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answerFind the circumference of a circle with a diameter of 13 meters. Use 3.14 as an approximation for E. Round your answer to thenearest whole meter. Enter only the numberThe solution isSubmitPassDon't know answerSave and close
Answers
Explanation:
The question wants us to obtain the circumference of the circle given that the diameter of the circle is 13 meters.
To do so, we will use the formula:
[tex]\begin{gathered} Circumference=\pi D \\ Where \\ \pi=3.14 \\ D=diameter=13\text{ meters} \end{gathered}[/tex]Therefore, the circumference will be
[tex]Circumference=3.14\times13=40.82\text{ }meters[/tex]Rounding off to the nearest whole number, we will have 41 meters
how do i find the type of relationship of a table? whether it is linear or quadradic and how do i find the formula for either relationship?
Answers
By finding the differences between dependent values, you can determine the degree of the model for data given as ordered pairs. If the first difference is the same value, the model will be linear. If the second difference is the same value, the model will be quadratic.
Also you can solve it by plotting the dots. If the graph seems a straight line it is linear and quadratic if it is a parabola.
For the data set given it is a linear relation
Line of best fit: y=3.18x+54.92
For x=7;
y=3.18*7+54.92
y=77.18
Find the ordered pairs for the x- and y-intercepts of the equation 8x - 2y = 16 and select the appropriate option below. The x-intercept is (-2, 0), the y-intercept is (0, 8). The x-intercept is (0, 2), the y-intercept is (-8, 0). The x-intercept is (2, 0), the y-intercept is (0, -8). The x-intercept is (0, -2), the y-intercept is (8, 0).
Answers
The given equation is
[tex]8x-2y=16[/tex]To find the x-intercept, we make y = 0.
[tex]\begin{gathered} 8x-2\cdot0=16 \\ 8x-0=16 \\ 8x=16 \\ x=\frac{16}{8} \\ x=2 \end{gathered}[/tex]Hence, the x-intercept is (2,0).To find the y-intercept, we make x = 0.
[tex]\begin{gathered} 8\cdot0-2y=16 \\ 0-2y=16 \\ -2y=16 \\ y=-\frac{16}{2} \\ y=-8 \end{gathered}[/tex]Hence, the y-intercept is (0,-8).Could you please help me with this? I need to solve it.
Answers
Given: ( -100 ) + ( -4 ) + ( -2 + 6) ( 3 )
Required: Evaluation
Explanation:
We shall here use BODMAS rule.
First solve the brackets and the addition and subtraction
[tex]\begin{gathered} (-100)+(-4)+(-2+6)(3) \\ =-100-4+(4)(3) \end{gathered}[/tex]Further,
[tex]\begin{gathered} =-100-4+12 \\ =-104+12 \end{gathered}[/tex]Solving
[tex]-104+12=-92[/tex]Final Answer:
[tex]-92[/tex]2. What is the equation of the line that passes through (5, 2) and isperpendicular to y =10x + 7?AC.y10x +yX+1021y = 10x - 4810B.-+52+52D.
Answers
Step 1
Given; What is the equation of the line that passes through (5, 2) and is
perpendicular to y =
10x + 7?
Step 2
The slope of the given line is;
[tex]\begin{gathered} m=10 \\ since,\text{ when we compare y=mx+b} \\ m=10 \end{gathered}[/tex]Slope of perpendicular lines have the following relationship;
[tex]\begin{gathered} m_1=-\frac{1}{m_2} \\ 10=-\frac{1}{m_2} \\ m_2=-\frac{1}{10} \end{gathered}[/tex]Therefore the required equation will be in the form of;
[tex]\begin{gathered} y=-\frac{1}{10}x+b \\ y=2 \\ x=5 \end{gathered}[/tex]Find b, the y-intercept
[tex]\begin{gathered} 2=-\frac{1}{10}(5)+b \\ 2=-\frac{1}{2}+b \\ 4=-1+2b \\ 2b=5 \\ b=\frac{5}{2} \end{gathered}[/tex]Thus the answer will be; Option B
[tex]\begin{gathered} y=-\frac{1}{10}x+\frac{5}{2} \\ \\ \\ \\ \end{gathered}[/tex]
Usually it takes Mrs. Manny 5.2 hours to grade her students' assignments. Thisweekend, her daughter Lexi is home and has offered to help. If it would take Lexi 6.4hours to grade the papers alone, how long will it take the two to finish the task of grading,working together?
Answers
Mrs. Manny needs 5.2 hours to grade her students' assignments. It would take Lexi 6.4 hours to grade the assignments alone.
In one hour, Mrs. Manny completes 1 / 5.2 = 0.1923 of the work.
In one hour, Lexi completes 1 / 6.4 = 0.15625 of the work
Together, they complete 0.1923 + 0.15625 = 0.34856 of the work.
The full work would take them 1 / 0.34856 = 2.87 hours.
It would take 2.87 hours for them to finish the task together.
The graphs of functions f(x) and g(x) = f(x) + k are shown below:g(x)65432f(x))3-3The value of k is.(1 point)
Answers
Solution
We know that :
g(x) = f(x) + k
For this case the answer is:
the value of k is: 4
A right triangle has the lengths of the legs are 60 centimeters and 80 centimeters. what is the length, in cm, of the hypotenuse?
Answers
The following image shows a diagram (not to scale) of the triangle with the indicated measurements:
We will label them as "a" and "b" for reference:
And we need to find the hypotenuse of the triangle, which is the side that is opposite to the 90° angle. We will label the hypotenuse as "c":
To solve the problem we have to us The Pythagorean Theorem:
[tex]c^2=a^2+b^2[/tex]Substituting the values of the legs a and b:
[tex]c^2=60^2+80^2[/tex]Since 60^2=3,600 and 80^2=6,400:
[tex]\begin{gathered} c^2=3,600+6,400 \\ c^2=10,000 \end{gathered}[/tex]Finally, to find the hypotenuse "c", take the square root of both sides of the equation:
[tex]\begin{gathered} \sqrt[]{c^2}=\sqrt[]{10,000} \\ c=\sqrt[]{10,000} \\ c=100 \end{gathered}[/tex]The length of the hypotenuse is 100 cm.
Answer: 100cm
Solve the following system using the substitution method. Enter your answer as an ordered pair in the form (x,y).3x - 2y = -95x + 10y = - 5
Answers
Given,
[tex]\begin{gathered} 3x-2y=-9\ldots\ldots\ldots(1) \\ 5x+10y=-5\ldots\ldots\text{.}(2) \end{gathered}[/tex]Multiply 1st equation by 5.
[tex]\begin{gathered} 5(3x-2y=-9) \\ 15x-10y=-45\ldots\ldots\text{.}(3) \end{gathered}[/tex]Solve equations (2) and (3)
[tex]\begin{gathered} 20x=-50 \\ x=\frac{-5}{2} \end{gathered}[/tex]Put x =-5/2 in equation (1)
[tex]\begin{gathered} 3x-2y=-9 \\ 3\times\frac{-5}{2}-2y=-9 \\ \frac{-15}{2}-2y=-9 \\ -2y=-9+\frac{15}{2} \end{gathered}[/tex]Further solved as,
[tex]\begin{gathered} -2y=\frac{-45+15}{5} \\ -2y=\frac{-30}{5} \\ -2y=-6 \\ y=3 \end{gathered}[/tex]Therefore, the value of x and y is -5/2 and 3.
i need help please with m and b#1 and graph
Answers
ANSWER
• m = 2
,• b = -5
EXPLANATION
This equation is written in slope-intercept form,
[tex]y=mx+b[/tex]Where m is the slope and b is the y-intercept.
In this case, the slope is m = 2 and the y-intercept is b = -5. This tells us that the line intersects the y-axis at y = -5, so that is our first point. Then, we can find the next point using the slope,
[tex]m=2=\frac{\Delta y}{\Delta x}[/tex]The next point is 1 unit to the right of the y-intercept, and 2 units up,
To draw a line we only need two points, so we have to draw a line passing through these two points.
I will share a photo of the question it is to complicated to right
Answers
Answer : 6
We are given the above fraction to be
[tex]\frac{3}{4}\text{ divided by }\frac{1}{8}[/tex][tex]\begin{gathered} To\text{ proc}eed\text{ with this expression, we n}eed\text{ to find the reciprocal of }\frac{1}{8} \\ \text{Hence, the reciprocal of }\frac{1}{8}\text{ is 8} \\ \frac{3}{4}\text{ x }\frac{8}{1} \\ =\text{ }\frac{3\text{ x 8}}{4} \\ =\text{ }\frac{24}{4} \\ =\text{ 6} \end{gathered}[/tex]The answer is 6
Before you can change a division operator to a multiplication operator, we need to find the reciprocal of the left hand side fraction
The fraction at the left hand side is 1/8
The reciprocal of 1/8 is 8
I need help with this math question all parts pleasePart 2: find y-interceptPart 3: find the zerosPart 4: Graph k(x)
Answers
Given the following function:
[tex]k(x)=x^3-5x^2[/tex]We will find the end behavior of the function.
the given function has a degree = 3 (odd)
And the leading coefficient is positive
the end behavior will be as follows:
[tex]\begin{gathered} x\to-\infty\Rightarrow k(x)\to-\infty \\ x\to\infty\Rightarrow k(x)\to\infty \end{gathered}[/tex]So, the answer will be:
The end behavior of the function is down to the left and up to the right.
===============================================================
Part (2), we will find the y-intercepts
The y-intercept is the value of y when x = 0
So, we will substitute x = 0 and then solve y
[tex]y=0^3-5(0^2)=0[/tex]So, the answer will be:
y-intercept = (0, 0)
================================================================
Part 3: we will find the zeros of k(x)
The zeros of the function are the values of x which make k(x) = 0
So, we will write the equation k(x) = 0 and then solve it for x.
[tex]\begin{gathered} x^3-5x^2=0 \\ x^2(x-5)=0 \\ x^2=0\to x=0 \\ x-5=0\to x=5 \end{gathered}[/tex]So, the answer will be:
Zeros of k: 0,5
===============================================================
Part 4: we will find the graph of k(x)
From the previous parts, we can conclude that
The graph of the function will be as shown in option D
jodie has 2 1/2 cases of soda to split between 5 families. what fraction of a case does each family receive?
Answers
We have in total 2 1/2 so we need to divide this in 5 equal parts
First we will convert our mixed number into a fraction
[tex]2\frac{1}{2}=\frac{4}{2}+\frac{1}{2}=\frac{5}{2}[/tex]Then we divide between 5
[tex]\frac{5}{2}÷\frac{5}{1}=\frac{5\ast1}{2\ast5}=\frac{1}{2}[/tex]ANSWER
1/2 case
у 10 8 P 4 2 0 2 6 8 10 Which ordered pair represents the location of point P on the coordinate plane? a (5,6) b (6,5) (6,7) d (7,6)
Answers
The ordered pair that represents the location of point P is (6,5)
Explanation:
In order to determine the position of point P, we will need to trace its position to the x axis and also trace its position to the y axis.
Tracing its position to the x axis, we find it corresponds to 6 units
Tracing its position to the y axis, we find it corresponds to 5 units.
This because each line represents 1 units. After the 4 units, the next line is 5 units.
Using the coordinates (x,y):
The ordered pair that represents the location of point P is (6,5)
accounts that earn 6% interest. If Emma’saccount earns simple interest and Paul’saccount earns compound interest, which is thevalue of each person’s account after 8 years?A. Emma – $2,960; Paul – $3,187.708. Emma – $960; Paul – $3,187.70C. Emma – $2,960; Paut- $ 1,187.70
Answers
So,
Remember that the simple interest of an initial amount after "t" years, can be found using the following formula:
[tex]A=P(1+rt)[/tex]Where A is the final amount, P is the initial amount, r is the rate and t are the years involved.
If we replace our values, Emma will has the following amount after 8 years:
[tex]\begin{gathered} A=2000(1+\frac{6}{100}(8)) \\ A=2960 \end{gathered}[/tex]So, Emma will has $2,960 after 8 years.
To find the amount that Paul will has, we should remember what the compound interest is.
Remember that the compound interest is given by the formula:
[tex]A=P(1+i)^n[/tex]Where A is the final amount, P is the initial amount, i is the rate and n are the years involved.
If we replace our values, Paul will has the following amount of money after 8 years:
[tex]\begin{gathered} A=2000(1+\frac{6}{100})^8 \\ A=3187.70 \end{gathered}[/tex]So, Paul will has $3187,70 after 8 years.
Therefore, the correct answer is A.