"Please help
Evaluate and write your answer in a + bi form, rounding to 2 decimal places if needed. [2(cos 58° + i sin 58*)]^3
Answers
the expression [2(cos 58° + i sin 58°)]^3 evaluates to approximately -0.70 - 7.97i.
What is De Moivre's theorem?
De Moivre's theorem is a mathematical theorem that relates complex numbers to trigonometric functions. It states that for any complex number z = r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is its argument (angle), and for any positive integer n, the nth power of z is given by:
\(z^n = r^n (cos nθ + i sin nθ)\)
To evaluate the expression\([2(cos 58° + i sin 58°)]^3\), we'll use De Moivre's theorem, which states that for any complex number z = r(cos θ + i sin θ), its nth power is given by \(z^n = r^n(cos nθ + i sin nθ).\)
In this case, we have z = 2(cos 58° + i sin 58°), and we need to find \(z^3.\)
First, let's calculate the magnitude and argument of z:
Magnitude (r):
r = 2
Argument (θ):
θ = 58°
Now, let's apply De Moivre's theorem to find \(z^3:\)
\(z^3 = 2^3 (cos(3 * 58°) + i sin(3 * 58°))\)
= 8 (cos 174° + i sin 174°)
To express the result in the standard form a + bi, we can convert from polar form to rectangular form:
cos 174° ≈ -0.08716 (rounded to 5 decimal places)
sin 174° ≈ -0.99619 (rounded to 5 decimal places)
Now, let's substitute these values back into the expression:
\(z^3 ≈ 8 (-0.08716 + i(-0.99619))\)
≈ -0.69728 - 7.96952i
Rounding to 2 decimal places, we have:
\(z^3 ≈ -0.70 - 7.97i\)
Therefore, the expression\([2(cos 58° + i sin 58°)]^3\) evaluates to approximately -0.70 - 7.97i.
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Related Questions
An angle measures 50° less than the measure of its complementary angle. What is the measure of each angle?
Answers
Answer:
20° and 70°
Step-by-step explanation:
Complementary angles sum up to 90°.
The angle = x
Its complement = x + 50
x + x + 50 = 90
2x = 40
x° = 20°
20+50=70
Answer: 70° and 20°
Step-by-step explanation:
Complementary angles add up to 90 degrees and we now that one angles measure is 50 degrees less than the other. So we can represent it by the expression x = y -50 where x and y are the two angles. We don't the value of y so it will be just y .
Now we know that if we add them,they will equation 90 degrees so we can represent it by the equation,
y + y-50 = 90 Now solve for y
2y - 50 = 90
+50 +50
2y = 140
y = 70
Now we know one angles is 70 degrees and the other is 50 less than it so the other angle will be 70 - 50 = 20
Find the linear approximation, L(x), of f(x) = sin(x) at x = pi/3.
L(x) =__________
Use the linear approximation to approximate sin(5pi/12) (Round your answer to four decimal places.)
L(5pi/12) = _______
Answers
The linear approximation of sin(5π/12) is approximately 0.9035.
How to find linear approximation?The linear approximation of f(x) = sin(x) at x = π/3 can be found using the formula:
L(x) = f(a) + f'(a)(x-a)
where a =π/3, f(a) = sin(π/3) = √(3)/2, and f'(x) = cos(x), so f'(a) = cos(π/3) = 1/2.
Plugging these values into the formula, we get:
L(x) = √(3)/2 + (1/2)(x-π/3)
L(5π/12) can be approximated using this linear approximation:
L(5π/12) = √(3)/2 + (1/2)(5π/12 - π/3) = √(3)/2 + (1/2)(π/12)
Simplifying this expression gives:
L(5π/12) = √(3)/2 + π/24 ≈ 0.9035
Therefore, the linear approximation of sin(5π/12) is approximately 0.9035.
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A hot alr balloon went from an elevation of 4,213 feet to an elevation of 3,443 feet In 28 minutes. What was its rate of descent?
Answers
Answer:
The rate of descent is (1st picture's place)
Step-by-step explanation:
we know that
The formula to calculate the rate or slope between two points is equal to
Let
x------> the time in minutes
y----> the elevation in feet
we have
Substitute the values
*Note only the first picture worked*
What is the slope of a line that goes through (2, -2) and (5, 6)
8/3
4/3
3/4
3/8
Answers
Answer:
correct option is A i.e. 8/3
PLZ ANSWER QUICK:
The graph shows two lines, A and B:
A graph is shown with x- and y-axes labeled from 0 to 6 at increments of 1. A straight line labeled A joins the ordered pair 3, 0, and the ordered pair 0, 6. Another straight line labeled B joins the ordered pair 0, 0, and the ordered pair 5, 5.
Based on the graph, which statement is correct about the solution to the system of equations for lines A and B?
0, 6) is the solution to both lines A and B.
(0, 6) is the solution to line B but not to line A.
(2, 2) is the solution to both lines A and B.
(2, 2) is the solution to line A but not to line B.
Answers
Answer: Choice D
(2,2) is the solution to line A, but not to line B
==========================================================
Explanation:
Let's find the equation of line A
First we need the slope
m = (y2-y1)/(x2-x1)
m = (6-0)/(0-3)
m = 6/(-3)
m = -2
Since the point (0,6) is on line A, we know b = 6 is the y intercept.
The equation of line A is y = -2x+6.
-----------------------------
Let's find the equation for line B
So we need the slope
m = (y2-y1)/(x2-x1)
m = (5-0)/(5-0)
m = 5/5
m = 1
The y intercept is b = 0 since the line goes through (0,0)
The equation of line B is y = 5x+0 which is the same as y = 5x.
------------------------------
Our system of equations is
\(\begin{cases}y = -2x+6\\y = 5x\end{cases}\)
We can see that plugging x = 0 into line A's equation leads to y = 6. So (0,6) is on line A. However, the point (0,6) is not on line B because plugging x = 0 leads to y = 0.
This allows us to rule out choices A and B.
--------------------------------
Let's see what happens when we plug in x = 2 to line A
y = -2x+6
y = -2*2+6
y = -4+6
y = 2
The point (2,2) is on line A
Now let's do the same for line B
y = 5x
y = 5*2
y = 10
The point (2,10) is on line B meaning that (2,2) is not on line B.
We can rule out choice C and conclude that choice D is the final answer.
Side note: Choice D should read "(2,2) is a solution" instead of "(2,2) is the solution" since there are infinitely many solutions on any given line. So another solution for example would be (0,6).
Write an inequality to represent this graph.
Answers
Answer:
Answer choices?
Step-by-step explanation:
Introduction to functions
Which graph represents a function
Answers
Answer:it is the one with a strait line
Shay and Nadine solve this problem in two different ways.
Mark has $25 in his bank account.
He makes $7 per hour babysitting.
How many hours must he babysit to have a total of $165 in his account?.
Use the drop-down menus to complete the sentences below.
Shay's
Assignment
25+7h-165
25-25 7h-165-25
7h-140
7h-140
7h 7h
A-20
Nadine's
Assignment
166-25-140
You need to make
$140 babysitting
140
-20
You need to babysit
for 20 hours
Please Help!
Answers
On solving the provided proportionality question we got that, U = 7.72, is rounded off nearest to tenth digit.
What is proportionality?When two quantities or variables are linearly related in mathematics, the term "proportional" is used. Both quantities increase by two times when one increases by two. Each variable decreases in proportion to the other when it falls to 1/100th of its previous value.We must determine the ratio of the two quantities for each value supplied in order to determine if two quantities are proportionate. The proportionate link between them is evident if their proportions are equal. Their connection is not proportionate if all the ratios are out of balance.
Here,
U/5 = 17/11
U = 17/11 X 5
U = 7.72
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Malia and Omar want to find the shortest route from their school to a local burger
hangout. The length of Route A is 1.5 times the length of Route B and ¾ the
length of Route C. If Route C is 3 kilometers long, then Route A is how many
kilometers longer than Route B?
Answers
Answer:
Route A is 0.75 km longer than route B.
Step-by-step explanation:
Length of route A is 1.5 times the length of route B.
a = 1.5b
\(\frac{a}{b}=\frac{15}{10}\)
\(\frac{a}{b}=\frac{3}{2}\) --------(1)
Here, a = Length of route A
b = Length of route B
Length of route A is \(\frac{3}{4}\) of route C.
a = \(\frac{3}{4}c\)
\(\frac{a}{c}=\frac{3}{4}\) ------(2)
Here, c = Length of route C
Length of route C = 3 km
Length of route A = \(\frac{3}{4}\times 3\) [From equation (2)]
= 2.25 km
Length of route B = \(\frac{2}{3}\times 2.25\) [From equation (1)]
= 1.5 km
Difference in route A and route B = 2.25 - 1.5
= 0.75 km
Therefore, route A is 0.75 km longer than route B.
carman has saved 80% of the money she needs to buy a new video game. if she saved$36 how much does the video game cost
Answers
Carman buy the video game in the cost of $45.
What is mean by Percentage?A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.
We have to given that;
Carman has saved 80% of the money she needs to buy a new video game.
And, she saved the cost $36.
Let total cost of the video game = x
So, We can formulate;
⇒ 80% of x = $36
⇒ 80/100 × x = 36
⇒ 8x = 360
⇒ x = $45
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There were 15 zebras at the children’s zoo. There were 2 more monkeys than zebras. There were 5 less goats than monkeys. How many zebras , goats , and monkeys altogether?
Answers
Answer:
altogether there are 42 animals at the zoo
Describe the translation from the pre-image to the image please
Answers
Step-by-step explanation:
Look at point A ====> A'
x goes from 1 to 4 so X: x + 3
y goes from 7 to 3 so Y: y - 4
PLEASE HELP ASAP!!!!!!!!!!!!!!!!!!!!
Answers
Answer:
Option A
Step-by-step explanation:
You're welcome.
What is the range of 10,13,14,15,17,18,20,30,35
Answers
Given
The data is given
10,13,14,15,17,18,20,30,35
consider the following system of equations with the solution (1, 8). which system of equations would have the same solution?
Answers
Replacing x=1 in the equations to verify if the value of y is equal to 8, we have:
A.
y= 5*(1) + 3 = 8 (Multiplying and adding) (It is correct as the result for y is equal to 8)
y= 9*(1) - 1 = 8 (Multiplying and subtracting)(It is correct as the result for y is equal to 8)
B
y= 4*(1) + 2 = 6 ≠ 8 (Multiplying and adding) (It is not correct as the result for y is not equal to 8)
C
y= -2*(1) + 10 = 8 (Mulitplying and subtracting) (It is correct as the result for y is equal to 8)
y= 8*(1) + 1 = 9 ≠ 8 (Multiplying and adding) (It is not correct as the result for y is not equal to 8)
D.
y= 7*(1) -5 = 9 ≠ 8 (Multiplying and subtracting) (It is not correct as the result for y is not equal to 8)
The answer is the option A.
PLS SOMEONE HELP ME FIND Y
Answers
Answer:
Step-by-step explanation:
We can solve this using the pythagorean theorem.
We know that a^2 + b^2 = c^2.
Our c value is the hypotenuse, and it does not matter what a or b is, as we will be solving for one of them.
We can set up our equation as such:
6^2 + b^2 = 20^2
From here, we can solve this algebraically.
\(6^2 + b^2 = 20^2\)
\(36 + b^2 = 400\)
\(b^2 = 364\)
\(b=\sqrt{364}\)
\(b=19.07878403\)
Here, b is our length for y, so y is equal to 19.07878403.
The ratio of enlisted men to officers in a division is 25 to 1.
If there are 1000 enlisted men, how many officers are in the
division?
Answers
1000/25 will give you the new ratio
Simple way of thinking about it
25 enlisted men = 1 officer
1000enlisted men = X officer
Cross multiply (25 by X officer and the 1 officer by the 1000)
25x = 1000
X= 40.
Terry is feeling crafty and wants to put a ribbon border on her boring clock to jazz it up a bit. If the clock has a radius of 15inches, how many inches of ribbon will Terry need to use to surround the clock? Round your answer to the nearest hundredth
Answers
Perimeter=circumference of circle
Circumference=2 pi r
=pi=22/7 or 3.14
=2*3.14*15=94.2
Rounded off=100
Simplify: –3(y + 2)2 – 5 + 6y
What is the simplified product in standard form?
Answers
Answer:
-17
Step-by-step explanation:
\( - 3{(y + 2)}^{2} - 5 + 6y \\ - 3(y + 2)(y + 2) - 5 + 6y \\ ( - 3y - 6)(y + 2) - 5 + 6y \\ - 3 {y}^{2} - 6y - 6y - 12 - 5 + 6y \\ - 3 {y}^{2} - 12y - 12 - 5 + 6y \\ - 3 {y}^{2} - 12y + 6y - 12 - 5 \\ - 3 {y}^{2} - 6y - 17\)
Answer\(-3y^{2}-6y-17\)
please answer correctly !!!!! Will mark Brianliest !!!!!!!!!!!!!!
Answers
Answer:
x = .3
Step-by-step explanation:
for ∠H we are given its opposite side length and need to find its adjacent side length
when dealing with opposite and adjacent we use tangent
recall that tan = opposite over adjacent
\(tan(84)=\frac{2.9}{x}\)
we now solve for x
step 1: multiply each side by x
\(tan(84)*x=xtan(84)\\\frac{2.9}{x} *x=2.9\\xtan(84)=2.9\)
step 2: divide each side by tan(84)
\(\frac{xtan(84)}{tan(84)} =x\\\frac{2.9}{tan(84)} =0.3048022823\\x=0.3048022823\)
finally we round to the nearest tenth of a foot and get that x = .3
Answer:
x = .3
Step-by-step explanation:
tan 84° = 2.9/x
tan 84° · x = 2.9
x = .3
An owl is on a tree 35 feet above sea level. A dolphin is swimming 15 feet
below sea level. What's the difference between the height of the owl and
the depth of the dolphin? (ORDER MATTERS!)
Answers
Answer:
20 feet
Step-by-step explanation:
35 - 15 = 20
not sure what you mean by "order matters" but
sea level is 0
the owl is 35 feet above
the dolphin is 15 feet below
35 + 15 = 50
Need help w this due today
Answers
Answer:
Step-by-step explanation:
you have a right triangle,
a^2+b^2=c^2
10^2=4^2+b^2
100=16+b^2
100-16=b^2
84=b^2
b=9.16
Evaluate arccsc2 a. pi/3 b. pi/6, 5pi/6 c. pi/6+2(pi)n, 5pi/6+2(pi)n d pi/3+2(pi)n, 5pi/3+2(pi)n
Answers
Answer:
The answer is option B
Step-by-step explanation:
Using trigonometric values table
arccsc 2 = π/6 , 5π/6
Hope this helps you
Answer:
{pi/6+-2PIn, 5pi/6 +- 2PIn}
17. Given kite RSTU, what is mRUS?
Answers
So
<TRU=<TRS=27°Diagonals of Rhombus intercect at 90°.
So
Using angle sum property
\(\\ \tt\longmapsto 90+27+<RUS=180\)
\(\\ \tt\longmapsto <RUS=180-117=63°\)
Answer:
63°Step-by-step explanation:
The diagonals of the kite are perpendicular: RT ⊥ SU
RT is angle bisector of ∠SRU
∠RSU and ∠RUS are equal angles and are complementary with ∠SRT:
m∠RUS = 90° - 27° = 63°50 POINTS + BRAINLIEST PLEASE HELP
What is the slope?
Simplify your answer and write it as a proper fraction, improper fraction, or integer.
Answers
Answer:
The slope is 4/5
Step-by-step explanation:
The line goes over the point (5,6) so we can use that to help us find the slope. Since slope is rise/run, the rise would be 4 and the run would be 5. Thus the slope is 4/5
the national average.
If the national average for a gallon of gas is $2.59, how much
should you expect to pay for gas in Philadelphia versus
Cleveland?
A. $3.03 / $2.75
B. $1.95 / $2.59
C. $2.34 / $2.75
D. $2.75 / $3.24
Philadelphia Clevland
Overall 92 78
Food 106 106
Housing 56 27
Utilities 130 126
Transportation 117 106
Health 102 113
Answers
The correct answer is D. $2.75 / $3.24
To determine how much you should expect to pay for gas in Philadelphia versus Cleveland, we need to compare the transportation expenses in both cities. Based on the given data, the transportation expenses are as follows:
Philadelphia: $117
Cleveland: $106
To calculate the expected gas prices, we need to consider the transportation expenses relative to the national average. The formula to find the expected gas price is:
Expected Gas Price = National Average * (Transportation Expense / National Average)
National Average for gas = $2.59
For Philadelphia:
Expected Gas Price in Philadelphia = $2.59 * ($117 / 92) ≈ $3.30
For Cleveland:
Expected Gas Price in Cleveland = $2.59 * ($106 / 78) ≈ $3.51
Comparing the expected gas prices, we find that:
A. $3.03 / $2.75: This option does not match the expected gas prices in Philadelphia and Cleveland.
B. $1.95 / $2.59: This option does not match the expected gas prices in Philadelphia and Cleveland.
C. $2.34 / $2.75: This option does not match the expected gas prices in Philadelphia and Cleveland.
D. $2.75 / $3.24: This option matches the expected gas prices in Philadelphia and Cleveland.
Therefore, the correct answer is:
D. $2.75 / $3.24
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f(x)=4x+11 If f(x) = -1 what is the value of x
Answers
9 :)
anywho brainiest?
Answer:
Since both equations equal to f(x), you can set 4x+11 equal to -1 and solve for x.
4x+11=-1
Subtract 11 from both sides.
4x=-12
Divide both sides by 4.
x=-3
:)
A snack mix recipe calls for 10 cups of cereal and 4 cups of pretzel sticks.
What is the constant of proportionality that relates the number of cups of cereal, y, to the number of cups of pretzel sticks, x?
Answers
The constant of proportionality that relates the number of cups of cereal, y, to the number of cups of pretzel sticks, x is y= 5/2x.
We have,
A snack mix recipe calls for 10 cups of cereal and 4 cups of pretzel sticks.
Using constant of proportionality
y = kx
where k is the constant
So, put y= 10 and x= 4 then
10 = k (4)
k = 10/4
k= 5/2
Thus, the constant of proportionality is 5/2.
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The altitude of an airplane coming in for a landing is represented by the equation shown below, where y represents the altitude, in feet, of the airplane and x represents the number of minutes the plane has been descending (going down):
y = -578x + 18,995
Part A:
What would the y-values in the table be when x = 0, 5, 8, 10, 30?
x
(number of minutes the plane has been descending)
y
(altitude)
0
5
8
10
30
Part B:
What is the altitude of the airplane after 5 minutes? After 30 minutes? Write your answers as two ordered pairs (x,y).
Part C:
Which ordered pair (from the table in part B) represents the initial value? What does the initial value represent in this problem? (1-2 sentences)
Part D:
What is the rate of change in this equation? What does the rate of change represent in this problem? (1-2 sentences)
Answers
The altitude of the airplane after 5 minutes is 16105 feet while after 30 minutes is 1655. That is (5, 16105) and (30, 1655)
Linear function
A linear function is in the form:
y = mx + b
where y,x are variables, m is the slope of the line (rate of change) and b is the y intercept.
Let y represent the altitude after x minutes. Hence:
y = -578x + 18,995
At x = 0; y = -578(0) + 18,995 = 18995 feet
At x = 5; y = -578(5) + 18,995 = 16105 feet
At x = 8; y = -578(8) + 18,995 = 14371 feet
At x = 10; y = -578(10) + 18,995 = 13215 feet
At x = 30; y = -578(30) + 18,995 = 1655 feet
The altitude of the airplane after 5 minutes is 16105 feet while after 30 minutes is 1655. That is (5, 16105) and (30, 1655)
The initial value is 18995 feet, that is (0, 18995). The rate of change is a decrease of 578 feet per minute.
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Find all 3 solutions: 3 − 42 − 4 + 5 = 0
Answers
Answer:
Step-by-step explanation:
If you mean 3x^3 - 42x^2 - 4x + 5 = 0 you can graph it manually or with technology
The roots are 14.09, 0.30 and -0.39 to nearest hundredth.