Answers
Answer:
C, [tex]4e^{i(7\pi/4)}[/tex]
Step-by-step explanation:
To remind you, Euler's formula gives a link between trigonometric and exponential functions in a very profound way:
[tex]e^{ix}=\cos{x}+i\sin{x}[/tex]
Given the complex number [tex]2\sqrt{2}-2i\sqrt{2}[/tex], we want to try to get it in the same form as the right side of Euler's formula. As things are, though, we're unable to, and the reason for that has to do with the fact that both the sine and cosine functions are bound between the values 1 and -1, and 2√2 and -2√2 both lie outside that range.
One thing we could try would be to factor out a 2 to reduce both of those terms, giving us the expression [tex]2(\sqrt{2}-i\sqrt{2})[/tex]
Still no good. √2 and -√2 are still greater than 1 and less than -1 respectively, so we'll have to reduce them a little more. With some clever thinking, you could factor out another 2, giving us the expression [tex]4\left(\frac{\sqrt{2}}{2} -i\frac{\sqrt{2}}{2}\right)[/tex] , and now we have something to work with.
Looking back at Euler's formula [tex]e^{ix}=\cos{x}+i\sin{x}[/tex], we can map our expression inside the parentheses to the one on the right side of the formula, giving us [tex]\cos{x}=\frac{\sqrt2}{2}[/tex] and [tex]\sin{x}=-\frac{\sqrt2}{2}[/tex], or equivalently:
[tex]\cos^{-1}{\frac{\sqrt2}{2} }=\sin^{-1}-\frac{\sqrt2}{2} =x[/tex]
At this point, we can look at the unit circle (attached) to see the angle satisfying these two values for sine and cosine is 7π/4, so [tex]x=\frac{7\pi}{4}[/tex], and we can finally replace our expression in parentheses with its exponential equivalent:
[tex]4\left(\frac{\sqrt2}{2}-i\frac{\sqrt2}{2}\right)=4e^{i(7\pi/4)}[/tex]
Which is c on the multiple choice section.
Related Questions
Ok I’m sorry I’m dumb but I really need help so if y’all can help me that would be amazing
Answers
Answer:
498
Step-by-step explanation:
38×12 + 7×6 = 498
don't worry!
Answer: B 498
Step-by-step explanation:12 times 38 because it is 12 eggs in each cartons plus 7 times 6 because she also bought 7 quail eggs equal to 498
a graphic designer chose a base font size and represented it as 1 on the scale. he then listed some consecutive scale sizes. two consecutive sizes were 5.832 and 10.4976.
what scale size came before 5.832 ?
Answers
Answer:
3.24
Step-by-step explanation:
You would use the two fonts given to find the scale factor.
So the scale factor would be 10.4976/5.832 = 1.8
So if the scale factor is 1.8 we can use that to find the font size just before that. To do this you would divide the font size by the scale factor.
So
5.832/1.8 = 3.24
The font size just before 5832 is 3.24.
Hope this helps, if there is an error in my work please correct me.
A counterexample for the expression sinx+cosx=1 is 45 degrees T or F
Answers
True
Step-by-step explanation:A counterexample is a special type of example that is an exception to a general rule, law, proposition or statement. Hence if we prove that [tex]sin(45^{\circ})+cos(45^{\circ})\neq 1[/tex] then this is a counterexample for the expression [tex]sin(x)+cos(x)=1[/tex]. Thus:
[tex]sin(x)+cos(x)=1 \\ \\ Substituting \ x=45^{\circ} \ we \ have: \\ \\ sin(45^{\circ})+cos(45^{\circ})=1 \\ \\\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}=1 \\ \\ Rewriting:\\ \\2\frac{\sqrt{2}}{2}=1 \\ \\ \therefore \sqrt{2}=1 \ FALSE![/tex]
So we observe that the equation is not fulfilled because [tex]\sqrt{2}\neq 1[/tex]. In conclusion:
A counterexample for the expression sin(x)+cos(x)=1 is 45° is true!
7. Jackie has one red pencil, one yellow pencil, one blue pencil,
one green pencil, one orange pencil, and one purple pencil. She
also has one red pen, one blue pen, and one black pen. If Jackie
randomly selects one pencil and one pen, what is the probability
that she will select a purple pencil and a black pen in that order?
Answers
Answer:
I believe it is 1/6 for the pencil and 1/3 for the pen. Altogether, 2/9.
Step-by-step explanation:
There is 1 purple pencil out of 6 and 1 black pen out of 3.
Using probability of independent events, it is found that there is a [tex]\frac{1}{15}[/tex] probability that she will select a purple pencil and a black pen in that order.
A probability is the number of desired outcomes divided by the number of total outcomes.
If two events, A and B, are independent, the probability of both happening is the multiplication of the probabilities of each happening, that is:
[tex]P(A \cap B) = P(A)P(B)[/tex]
In this question:
The pencil and the pen are independent.One of the five pencils are purple, thus [tex]P(A) = \frac{1}{5}[/tex].One of the three pens are black, thus [tex]P(B) = \frac{1}{3}[/tex].The probability is:
[tex]P(A \cap B) = P(A)P(B) = \frac{1}{5} \times \frac{1}{3} = \frac{1}{15}[/tex]
[tex]\frac{1}{15}[/tex] probability that she will select a purple pencil and a black pen in that order.
A similar problem is given at https://brainly.com/question/23855473
The theoretical probability of an event occurring is 2/5
which best describes the experimental probability associated with this
A)Out of every 5 trials, the desired outcome will occur exactly 2 times
B)Out of every 5 trials the desired outcome will occur approximately 2 times
C)Out of every 7 trials, there will be 2 desired outcomes and 5 undesired outcomes
D)
Out of every 7 trials, there will be 5 desired outcomes and 2 undesired outcomes
Answers
Answer:
The answer is B. Hope this helped! c;
Step-by-step explanation:
Answer:
B)Out of every 5 trials the desired outcome will occur approximately 2 times
Step-by-step explanation:
The theoretical probability of an event occurring is 2/5. Which best describes the experimental probability associated with this?
The answer is -
Out of every 5 trials, the desired out come will be approximately 2 times.
Theoretical probability is based on reasoning. It is written as a ratio of the number of favorable outcomes to the number of possible outcomes.
Need help please.
Chantale wants to make a graph of her walk home from school yesterday.
A. She walked toward home from school.
B. She ran back to school to get a notebook she forgot.
C. She spent 2 minutes at school getting her notebook.
D. She walked all the way back home.
Drag a segment to each box to show the shape of the graph for that interval.
Answers
Answer:
See diagram below
Step-by-step explanation:
I added the picture but not sure if you can see it? Hope it helps!
Final answer:
Chantale's motion can be graphically represented on a position-time graph, with walking segments shown by lines with positive slopes, the running back segment by a line with a negative slope, and the stationary segment at school by a horizontal line.
Explanation:
The question pertains to plotting a graph of Chantale's motion as she moves to and from school, which is a concept from kinematics in physics. We can visualize her walk as a position-time graph, where each part of her travel (A to D) corresponds to a different segment of the graph.
A. Walking towards home: This would be represented by a line moving away from the origin (assuming the school is the origin) with a positive slope, showing that distance from school is increasing over time.
B. Running back to school: This segment of the graph will have a negative slope as Chantale moves closer to the origin (school).
C. Spending 2 minutes at school: During this time, her position does not change, so the graph will show a flat, horizontal line indicating no movement.
D. Walking back home: This would again be a line with a positive slope as she moves away from the school to her house.
To encompass the entire journey, you would see a graph with segments that reflect these motions, and each segment would follow the order of actions A to D.
Which of the following represents the factored form of f(x) = x3 − 81x? f(x) = x(x − 9)2 f(x) = (x + 9)(x − 9) f(x) = x(x + 9)(x − 9) f(x) = x(x2 − 9)
Answers
Answer:
f(x) = x(x + 9)(x − 9)
Step-by-step explanation:
Answer: f(x) = x(x+9)(x-9)
Step-by-step explanation:
CAN SOMEONE HELP ME SOLVE THIS
Answers
Answer:
60
Step-by-step explanation:
Answer:
x = 60°
Step-by-step explanation:
The sum of the 3 angles in a triangle = 180°, thus
x + x + x = 180
3x = 180 ( divide both sides by 3 )
x = 60
The triangle is equilateral
How many desserts are available?
Answers
4 desserts
The number of meals can be expressed by c * s * d = m, where c is main courses, s is salads, d is desserts, and m is meals.
Fill in the variables.
6 * 2 * d = 48
Simplify.
12 * d = 48
Divide both sides by 12.
d = 4
So, there are 4 desserts.
Solve the inequality.
x + 11 > 2
Answers
Let's solve your inequality step-by-step.
x+11>2
Step 1: Subtract 11 from both sides.
x+11−11>2−11
x>−9
Answer:
x>−9
What is the absolute value of
|-14|
|-16|
Answers
Answer:
14 and 16
Step-by-step explanation:
absolute value of a negative is the positive
Answer:
Answer is 14 and 16.
Step-by-step explanation:
Absolute value is never negative.
Hope my answer has helped you!
A company dyes two sizes of rugs. A small rug requires 8 hours for dyeing and a medium-size rug requires 12 hours for dyeing. The dyers have to make at least 22 rugs, and they must do it in less than 240 hours. Let x equal small rugs and y equal medium rugs. Which of the following inequalities can be paired with x + y ≥ 22 to create a system that represents this situation?
8x + 12y > 240
12x + 8y > 240
8x + 12y < 240
12x + 8y < 240
Answers
Answer:
8x + 12y < 240
Step-by-step explanation:
If x equals the number of small rugs and y equals the number of medium rugs dyed, we would need to multiply both variables by the number of hours taken to dye each sized rug. It takes 8 hours to dye a small rug (8x), and 12 hours to dye a medium rug (12y).
Then, to find the total number of hours they dye them in, add the terms together.
8x + 12y
The company must dye the rugs in less than 240 hours, so we will use a 'less than' (inequality) sign (it cannot be 240, or any more than 240).
8x + 12y < 240
Answer:
c
Step-by-step explanation:
In this triangle, what is the value of x?
Enter your answer, rounded to the nearest tenth, in the box.
x = ____ km
Answers
Answer: [tex]x=56.5\ km[/tex]
Step-by-step explanation:
Given the right triangle in the image, you need to remember the following identity:
[tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex]
Observe the triangle. You can identify that:
[tex]\alpha=28\°\\adjacent=x\\hypotenuse=64[/tex]
Then, knowing these values, you can substitute them into [tex]cos\alpha=\frac{adjacent}{hypotenuse}[/tex]:
[tex]cos(28\°)=\frac{x}{64}[/tex]
Finally, you have to solve for "x".
Therefore, the value of "x" rounded to the nearest tenth is:
[tex]64*cos(28\°)=x\\x=56.5\ km[/tex]
018-2019) Q4
Describing the structure on the TTO
HE
Multiplying a trinomial by a trinomial follows the same
steps as multiplying a binomial by a trinomial. Determine
the degree and maximum possible number of terms for
the product of these trinomials: (x2 + x + 2)(x2 – 2x + 3).
Explain how you arrived at your answer.
Answers
Answer:
[tex]\boxed{\text{Degree = 4; Max. no. of terms = 5}}[/tex]
Step-by-step explanation:
(x² + x + 2)(x² – 2x + 3)
1. Degree of product
The degree of the product of two polynomials is the sum of their degrees.
Each polynomial is of degree 2, so the degree of the product is 4.
That is, the term of highest degree must come from x² × x² = x⁴.
2. Maximum number of terms
The maximum number of terms is one more than the degree of the product.
Thus, there is a maximum of five terms in the product: one for each power of x plus a constant term.
P = a₁x⁴ + a₂x³ + a₃x² + a₄x + a₅
please help i will give brainliest!
Which statement is best represented by the inequality p>48?
1.The basketball team scored less than 48 points.
2.The basketball team scored 48 fewer points than last year’s team.
3.The basketball team scored greater than 48 points.
4.The basketball team scored exactly 48 points.
Answers
Answer:
2.
Step-by-step explanation:
The reason I say this one is because that equation say p is greater than 48 that is why,
Answer: Option number 3!
Find the volume of the cone with the given dimensions: d = 26 cm, h = 10 cm
Answers
Answer:
¹⁶⁹⁰/₃π cm³ ≈ 1769.76 cm³
Step-by-step explanation:
Equation of a volume of a cone: V = ¹/₃ * πr²h
1. Find r (radius): r = d/2 = 26/2 = 13 cm
2. Plug in everything into equation: V = ¹/₃ * π(13)² * 10
3. Square: V = ¹/₃ * π * 169 * 10
4. Multiply: V = ¹⁶⁹⁰/₃π cm³ ≈ 1769.76 cm³
if it has a diameter of 26 cm, then its radius is half that, or 13.
[tex]\bf \textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\ \cline{1-1} r=13\\ h=10 \end{cases}\implies V=\cfrac{\pi (13)^2(10)}{3} \\\\\\ V=\cfrac{1690\pi }{3}\implies V\approx 1769.76[/tex]
How does the graph of g(x)=3x-2 compare to the graph of f(x)=3x
Answers
Answer:
Step-by-step explanation:
the graph of f(x)=3x becomes the graph of g(x)=3x-2 if the entire graph of f(x) = 3x is translated down by 2 units.
an electric motor makes 3000 revolution per minutes how many degrees does it rotate in a second
Answers
Answer: 18,000deg
Step-by-step explanation:
(3000rev/min)*(1min/60s)*(360deg/1rev)=18,000deg
Answer:
18,000
Step-by-step explanation:
We know that the electric motor makes 3000 revolutions in a minute/60 sec.
In one second, the motor makes 3000/60 revolutions.
In one second, the motor makes 50 revolutions.
∴ 1 revolution = 360 degrees
50 revolutions = 360 × 50 degrees
50 revolutions = 18,000 degrees
Graph the line Y=2/3x-1
Answers
Since this is a linear question in y=mx+b form, and where m is the slope of the line and b is the y-intercept....
The slope in this problem is 2/3
And y intercept is -1
So you would graph the point -1 on y axis then go up 2/3 since it’s positive( go up 2 over 3)
Brainliest + Points
Can someone please help?
Answers
See the attached picture.
The Y values would remain the same, the X values become the negative of the original values.
Which graph represents a reflection of f(x) = 2(0.4)x across the y-axis?
Answers
Answer:
The graph in the attached figure
Step-by-step explanation:
we know that
In a reflection across the y-axis the y-coordinate remains the same, but the x-coordinate is transformed into its opposite
we have
[tex]f(x)=2(0.4)^{x}[/tex]
The reflection of f(x) across the y-axis is equal to the function g(x)
[tex]g(x)=2(0.4)^{-x}[/tex]
The graph in the attached figure
The graph which best represents a reflection of f(x) across the y-axis is; f(x) = 2(0.4)^-x.
[tex]f(x) = 2(0.4)^{-x}[/tex]
What is the reflection of the graph of the function?A reflection of a point over the y -axis is given by a rule in accordance with mathematical convention. The rule in discuss for a reflection over the y -axis is (x,y)→(−x,y).
Hence, the reflection of the graph is given by the function; f(x) = 2(0.4)^-x.
[tex]f(x) = 2(0.4)^{-x}[/tex]
Read more on reflection of a graph;
https://brainly.com/question/4335018
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Two cars are 400 miles apart when they start moving towards each other. The speed of the first car is 30 mph. The second car moves 20mph faster than the first car. After how many our will the cars meet
Answers
Answer:
5 hr.
Step-by-step explanation:
this is because one car is moving at 30 miles an hour and the other at 30+20 miles an hour = 50mph. so after 1 hour one car would have gone 30 miles and the other 50 = 80 altogether not 400 yet. After 2 hours one car would have gone 60 miles and the other 100 = 160 not 400 yet. ECT... After five hours one car has gone 150 mile (30mph*5hr = 150miles) and the other 250miles (59mph*50hr = 250miles) = 400, since 150+250=400. Therefore it would take 5hr for the cars to meet. and crash ._.
Final answer:
The two cars traveling towards each other, one at 30 mph and the other at 50 mph, will meet after 5 hours, since their combined speed is 80 mph and they are 400 miles apart.
Explanation:
To solve the problem of when two cars moving towards each other will meet, we can use the concept of relative speed. The first car travels at 30 mph, and the second car travels at the faster speed of 50 mph (since it is 20 mph faster than the first car). The combined speed at which the two cars are moving towards each other is the sum of their individual speeds.
The formula for time is:
Time = Distance / Speed
Given that the two cars are 400 miles apart, and the combined speed is 30 mph + 50 mph = 80 mph, the time it takes for the two cars to meet is:
Time = 400 miles / 80 mph = 5 hours
Therefore, the cars will meet after 5 hours of travel.
On monday ,248 students went on a trip to the zoo.All 4 buses were filled and 8 students had to travel in in cars. How many students were in each bus?
Answers
Answer:
60 students were in each bus.
Step-by-step explanation:
248-8=240
240 students rode on busses.
240/4= 60
60 students per bus.
248-8= 240 (taking away 8 because 8 of the students are not taking the bus)
240/4=60 (dividing the number we got by 4 because there are 4 busses)
i hope i helped :)
What are the zeros of the function f(x)=x^3+x^2-6x
PLEASE HELP ASAP!!
Answers
Answer:
A
Step-by-step explanation:
Given
x³ +x ² - 6x ( to find the zeros equate to zero ), that is
x³ + x² - 6x = 0 ← factor out x from each term
x(x² + x - 6) = 0
To factorise the quadratic
Consider the factors of the constant term (- 6) which sum to give the coefficient of the x- term (+ 1)
The factors are + 3 and - 2, since
3 × - 2 = - 6 and 3 - 2 = + 1, hence
x(x + 3)(x - 2) = 0
Equate each factor to zero and solve for x
x = 0
x + 3 = 0 ⇒ x = - 3
x - 2 = 0 ⇒ x = 2
The zeros are x = - 3, x = 0, x = 2 → A
If the endpoints of have the coordinates A(9, 8) and B(-1, -2), what is the midpoint of AB?
A.
(5, 3)
B.
(4, 5)
C.
(5, 5)
D.
(4, 3)
Answers
For this case we have that by definition, the formula of the midpoint is given by:
[tex]MP = (\frac {x_ {1} + x_ {2}} {2}, \frac {y_ {1} + y_ {2}} {2})[/tex]
We have the points are:
[tex](x_ {1}, y_ {1}) :( 9,8)\\(x_ {2}, y_ {2}): (- 1, -2)\\MP = (\frac {9 + (- 1)} {2}, \frac {8 + (- 2)} {2})\\MP = (\frac {9-1} {2}, \frac {8-2} {2})\\MP = (\frac {8} {2}, \frac {6} {2})\\MP = (4,3)[/tex]
Answer:
Option D
Answer:
option D is right on plato
Step-by-step explanation:
Use DeMoivre's Theorem to write the complex number in standard form.
Answers
Answer:
A
Step-by-step explanation:
First express [tex]\sqrt{2}[/tex] + i[tex]\sqrt{2}[/tex] in trig form
| z | = [tex]\sqrt{(\sqrt{2})^2+(\sqrt{2})^2 }[/tex] = [tex]\sqrt{4}[/tex] = 2
Θ = [tex]tan^{-1}[/tex]( [tex]\frac{\sqrt{2} }{\sqrt{2} }[/tex]) = [tex]tan^{-1}[/tex]( 1) = [tex]\frac{\pi }{4}[/tex]
Thus
[tex]\sqrt{2}[/tex] + i[tex]\sqrt{2}[/tex] = 2 [ cos[tex]\frac{\pi }{4}[/tex] + isin[tex]\frac{\pi }{4}[/tex] ]
Hence
([tex]\sqrt{2}[/tex] + i[tex]\sqrt{2}[/tex] )³
= 2³ [ cos ([tex]\frac{\pi }{4}[/tex] × 3) + isin([tex]\frac{\pi }{4}[/tex] × 3 ) ]
= 8 [ cos([tex]\frac{3\pi }{4}[/tex] ) + isin([tex]\frac{3\pi }{4}[/tex]) ] → A
The standard form of the give complex number is
8 [ cos(\frac{3\pi }{4} ) + isin(\frac{3\pi }{4}) ]
We have given that,
First express [tex]\sqrt{2} + i\sqrt{2}[/tex] in trig form
[tex]| z | = \sqrt{(\sqrt{2})^2+(\sqrt{2})^2 } = \sqrt{4} = 2[/tex]
[tex]\theta = tan^{-1}( \frac{\sqrt{2} }{\sqrt{2} }) = tan^{-1}( 1) = \frac{\pi }{4}[/tex]
Thus, [tex]\sqrt{2} + i\sqrt{2} = 2 [ cos\frac{\pi }{4} + isin\frac{\pi }{4} ][/tex]
What is the Demoivre's theore?[ r(cos θ + i sin θ) ]^n = r^n(cos nθ + i sin nθ)
Hence, [tex](\sqrt{2} + i\sqrt{2} )^3[/tex]
[tex]= 2^3 [ cos (\frac{\pi }{4} \times 3) + isin(\frac{\pi }{4} \times 3 ) ][/tex]
[tex]= 8 [ cos(\frac{3\pi }{4} ) + isin(\frac{3\pi }{4}) ] \implies A[/tex]
Thererefore the standard form of the give complex number is
8 [ cos(\frac{3\pi }{4} ) + isin(\frac{3\pi }{4}) ].
To learn more about the complex number visit:
https://brainly.com/question/10662770
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Someone please help me!
Answers
Answer:
ΔAGB ≅ ΔCGBStep-by-step explanation:
Look at the picture
A teacher divided the class into two groups of equal size. 3/5 of the first group are right-handed. 80% of the second group are right-handed.
What fraction of the class is right handed??
Answers
Answer:
7/10
Step-by-step explanation:
To make things simple, let's turn 80% into a fraction, 8/10. To get a common denominator let's turn 8/10 into 4/5. Since they are both of equal size, the first group would have 2/5 left-handed people and the second 1/5 left-handed people. Let's imagine that each group has 5 people in it. There would be 10 people in total. There is 1 person for each fifth so we can add now. 4 + 3 = 7 right handed people. To check, 2 + 1 = 3 and 3 + 7 = 10. so 7/10 of the class is right-handed.
The fraction of the class is right handed is 7/10
We have given that
80% of the second group are right-handed.
let's turn 80% into a fraction,
What is the fraction form of 80%?The fraction form of 80% can be calculated by dividing 80 by 100
So we get,
[tex]=\frac{80}{100} \\\\=\frac{8}{10}[/tex]
=8/10.
To get a common denominator let's turn
8/10 =4/5 .....(divide the numerator and denominator by 2)
Since they are both of equal size,
The first group would have 2/5 left-handed people and
The second 1/5 left-handed people.
Let's imagine that each group has 5 people in it.
There would be 10 people in total.
There is 1 person for each fifth so we can add now.
4 + 3 = 7 .....(1)
right handed people.
we can check it,
2 + 1 = 3 and 3 + 7 = 10. .........(2)
Therefore we get,
7/10 of the class is right-handed.
The fraction of the class is right handed is 7/10
To learn more about the fraction visit:
https://brainly.com/question/78672
a box has a length of 12 inches and a width of 10 inches . If the volume of the box is 960 cubic inches. What is it's height?
Answers
length* width
12*10= 120
Volume /120
960/120=8
height is 8 inches
Two numbers are of the ratio 3:5. If their difference is 18, find the numbers.
Answers
[tex]x \div y = 3 \div 5 \\ it \: means = \\ 3y = 5x \\ x - y = 18 \\ [/tex]
To solve we multiply the last equation by 5 and we get the following:
5x-5y=90
But 5x=3y and we replace it in this equation:
3y-5y=90
-2y=90
y=-45
and x-(-45)=18 , x+45=18
x=-27