[30 PTS!!] Using mathematically precise language, explain in detail how you would multiply the complex number [tex]z_1=r_1(cos[/tex]θ[tex]_1+i sin[/tex]θ[tex]_1)[/tex] with the complex number [tex]z_2=r_2(cos[/tex]θ[tex]_2+isin[/tex]θ[tex]_2)[/tex].
![[30 PTS!!] Using Mathematically Precise Language, Explain In Detail How You Would Multiply The Complex](https://brainimage.s3.us-east-005.backblazeb2.com/contents/attachments/MVgV0cSVSrSvGz7khy4KfjpdadXwVO52.png)
Answers
Multiplying z₁ and z₂ involves calculating their products
The product of z₁ and z₂ is r₁r₂(cos(Θ₁ + Θ₂) + isin(Θ₁ + Θ₂))
How to multiply the complex numbers?The numbers are given as:
z₁ = r₁(cosΘ₁ + isinΘ₁)
z₂ = r₂(cosΘ₂ + isinΘ₂)
The product is represented as:
z₁ * z₂ = r₁(cosΘ₁ + isinΘ₁) * r₂(cosΘ₂ + isinΘ₂)
This gives
z₁ * z₂ = r₁r₂(cosΘ₁ + isinΘ₁) (cosΘ₂ + isinΘ₂)
Expand
z₁ * z₂ = r₁r₂(cosΘ₁cosΘ₂ + icosΘ₁sinΘ₂+ isinΘ₁cosΘ₂ + i²sinΘ₁sinΘ₂ )
In complex numbers,
i² = -1.
So, we have:
z₁ * z₂ = r₁r₂(cosΘ₁cosΘ₂ + icosΘ₁sinΘ₂+ isinΘ₁cosΘ₂ - sinΘ₁sinΘ₂ )
Rewrite as:
z₁ * z₂ = r₁r₂(cosΘ₁cosΘ₂ - sinΘ₁sinΘ₂ + icosΘ₁sinΘ₂+ isinΘ₁cosΘ₂ )
Apply sine and cosine ratios
z₁ * z₂ = r₁r₂(cos(Θ₁ + Θ₂) + isin(Θ₁ + Θ₂))
Hence, the product of z₁ and z₂ is r₁r₂(cos(Θ₁ + Θ₂) + isin(Θ₁ + Θ₂))
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Related Questions
Help on number 8 please!
Answers
Answer:
ab/3
Step-by-step explanation:
I don't really know how to explain.
Μα†hway
hope this helps!
= (-1/-3)(a2/a)(b2/b)(c2/c2)
=(1/3)[a^(2-1)][b^(2-1)][c^(2-2)]
= (1/3)(a)(b)(1)
=ab/3
Answer = ab/3
A banner is made of a square and a semicircle. The square has side lengths of 26 inches. One side of the square is also the diameter of the semicircle. What is the total area of the banner? Use 3.14 for π.
Answers
Answer:
941.33 in^2.
Step-by-step explanation:
The area of a semicircle = 1/2 area of circle = 1/2 * 3.14 * radius^2.
The radius = 1/2 * 26 = 13 in.
Total area = area of square + area of semicirccle
= 26^2 + 1/2 * 3.14 * 13^2
= 676 + 265.33
= 941.33 in^2.
Im so desperate
ms. davis sold $309.00 of books. at the end of the day, she earned $27.81 in commission for those sales. what is ms. davis' commission rate?
question 8 options:
11%
9%
3.4%
2.81%
Answers
so Ms. Davis gets hmmm say "x" percent from sales as her commission, and we know she did in sales $309 and made $27.81.
if we take 309 to be the 100%, and x% of that is 27.81, what's "x"?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 309 & 100\\ 27.81& x \end{array} \implies \cfrac{309}{27.81}=\cfrac{100}{x} \\\\\\ 309x=2781\implies x=\cfrac{2781}{309}\implies x=9[/tex]
What are the domain, range, and asymptote of h(x) = (1.4)x + 5?
domain: {x | x is a real number}; range: {y | y > 5}; asymptote: y = 5
domain: {x | x > 5}; range: {y | y is a real number}; asymptote: y = 5
domain: {x | x > –5}; range: {y | y is a real number}; asymptote: y = –5
domain: {x | x is a real number}; range: {y | y > –5}; asymptote: y = –5
Answers
Answer:
The answer is
→ A.) Domain: {x | x is a real number}; range: {y | y > 5}; asymptote: y = 5
Step-by-step explanation:
For every real x, (1.4)x is always positive, therefore the range would be y > 0, but (1.4)x + 5 adds 5 to every integer in the original range. This suggests that the real range is y | y > 5, or all positive values bigger than 5.
This is the sole choice for the range that A has, therefore, it should. (The other two characteristics hold true, as can be any real integer and lim h(x) = 5 x, implying that y = 5 is a horizontal asymptote of h(x).)
Therefore, A.) Domain: {x | x is a real number}; range: {y | y > 5}; asymptote: y = 5 is your answer.
Hope this helped! :^)
⭐Brainliest will be appreciated!⭐
✉️Message me if you have any more questions!✉️
- ✨7272033Alt✨
Submitted on 4/20/2022 at 1:37 PM
(To avoid plagarism, by users whom decide to copy.)
Domain: {x | x is a real number}; range: {y | y > 5}; asymptote: y = 5. Option A is correct.
The given function is [tex]h(x)=1.4^x+5[/tex].
The domain of a function is the set of all valid inputs (x-values) for which the function is defined.
In this case, there are no restrictions on the base (1.4) raised to any real exponent, and thus, the domain includes all real numbers.
Therefore, Domain: {x | x is a real number}
The range of a function is the set of all possible output (y-values) that the function can produce.
In this case, the base (1.4) raised to any real exponent will always be positive, and when you add 5 to a positive number, you get values greater than 5.
Hence, the range consists of all real numbers greater than 5.
Therefore, Range: {y | y > 5}.
An asymptote is a line that a function approaches but never crosses.
For this exponential function, there are no vertical asymptotes since it is defined for all real numbers.
However, there is a horizontal asymptote. As the value of x approaches negative or positive infinity, the function h(x) will approach 5.
Therefore, Asymptote is y = 5.
Hence, option A is correct, domain: {x | x is a real number}; range: {y | y > 5}; asymptote: y = 5.
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May you help me please I am stressed and struggling
Answers
Answer:
The answer would be 2x+4+3x+4, 5x+8, and 2(2x+4) +x
Step-by-step explanation:
Step 1. Simplify 2x+4+3x+4 into 5x (combine the x's) +8 (combine the 2 4's)
Step 2. Simplify 2(2x+4)= x into 4x + 8 + x (<-- distribute the 2 to the 2x and 4) and 4x+x would equal 5x meaning 5x + 8.
I hope that this helps! :)
Question 4
Which shows the solution set to the
inequality below?
2x + 35-5
Answers
Answer:x=-15
Step-by-step explanation:
2x+35-5=0
-35 +5. -35 +5
2x=-30
/2 /2
X=-15
A parking lot occupies a rectangular space measuring 120m by 80m. The owner wants to increase the parking lot by 25%. What would be the new dimension of the parking lot?
Answers
Answer:
150metres by 100metres
i need help 9th grade math
Answers
Answer:
(f + g)(x) = x + 5
Explanation:
f(x) = 2x + 3
g(x) = 3x + 2
===========
(f + g)(x)f(x) + g(x)2x + 3 + 3x + 25x + 5Answer:
[tex](f+g)(x)=5x+5[/tex]
Step-by-step explanation:
Given:
[tex]f(x)=2x+3[/tex]
[tex]g(x)=3x+2[/tex]
[tex]\begin{aligned}(f+g)(x) & =f(x)+g(x)\\ & = (2x+3)+(3x+2)\\ & = 2x+3+3x+2\\ & =2x+3x+3+2\\ & = 5x + 5\end{aligned}[/tex]
Graph x≥2.
look at attachment
Answers
Match the vocabulary word with the correct definition.
Answers
Answer:
A. Variable
B. Unit Rate
C. Inverse Operations
D.System of linear equations
what is the value of the expression below 2816 * 7
Answers
Answer:
2816 x 7 is 19712.
Step-by-step explanation:
I'm not going into detail, but simply, use a calculator
Answer:
the answer is 19,712
Step-by-step explanation:
ur welcome
Help please, keep going down on ixl and it’s due in the morning, currently 1:33 am
Answers
Answer:
82 yd²
area of rectangle:
Length * Width
6 * 424 yd²area of 2nd rectangle
5 * 210 yd²area of triangle
1/2 * base * height1/2 * 12 * 848 yd²total:
18 yd² + 10 yd² + 24 yd²82 yd²What is the measure of angle 8?
Answers
Answer:
32
Step-by-step explanation:
First, let's find angle 6 as we will need that to find angle 8.
angle 6 = 180 - 122 = 58(straight line angles equal to 180)
now sum of angles in a triangle = 180
so angle 8 = 180 - 90 - 58
= 32
Hope it helps:)
Answer:
32°
Step-by-step explanation:
We know that the exterior angle of the triangle is equal to the sum of the interior opposite angles.
Therefore,
< 1 + < 8 = 122°
Also, given that,
<1 = 90°
So,
< 1 + < 8 = 122°
90° + < 8 = 122
< 8 = 32°
A contractor records the areas, in square feet, of a small sample of houses in a neighborhood to determine data about the neighborhood. they are: 2,400; 1,750; 1,900; 2,500; 2,250; 2,100 which of the following represents the numerator in the calculation of variance and standard deviation? (225)2 (–425)2 (–275)2 (325)2 (75)2 (–75)2 = 423,750 (650)2 (–150)2 (–600)2 (250)2 (150)2 (–300)2 = 980,000 (250)2 (–400)2 (–250)2 (350)2 (100)2 (–50)2 = 420,000 what is the variance? what is the standard deviation, rounded to the nearest whole number?
Answers
Option c is the correct answer which represents the numerator in the calculation of variance and standard deviation.
What are the variance and standard deviation?In statistics, variance is a measure of the variation of data from the mean, whereas standard deviation is the measure of the normal distribution of statistical data.
A contractor records the areas, in square feet, of a small sample of houses in a neighborhood to determine data about the neighborhood.
2,400; 1,750; 1,900; 2,500; 2,250; 2,100
we need to find which of the following represents the numerator in the calculation of variance and standard deviation.
(225[tex])^{2}[/tex] + (–425[tex])^{2}[/tex]+ (–275[tex])^{2}[/tex] + (325[tex])^{2}[/tex] + (75[tex])^{2}[/tex]+ (–75[tex])^{2}[/tex] = 423,750
(650[tex])^{2}[/tex] + (–150[tex])^{2}[/tex] + (–600[tex])^{2}[/tex] + (250[tex])^{2}[/tex] + (150[tex])^{2}[/tex] + (–300[tex])^{2}[/tex] = 980,000
(250[tex])^{2}[/tex] + (–400[tex])^{2}[/tex] + (–250[tex])^{2}[/tex] + (350[tex])^{2}[/tex] + (100[tex])^{2}[/tex] + (–50[tex])^{2}[/tex] = 420,000
92416.60
The variance is 84000
The standard deviation is 290
mean = (2400+1750+1900+2500+2250+2100)/6
mean = 2150
Subtract the data values from the mean to get
2400-2150 = 250
1750-2150 = -400
1900-2150 = -250
2500-2150 = 350
2250-2150 = 100
2100-2150 = -50
The differences are 250, -400, -250, 350, 100, -50
Then you square those values and add up the squares
(250)^2 + (-400)^2 + (-250)^2 + (350)^2 + (100)^2 + (-50)^2 = 420,000
Hence, option c is the correct answer which represents the numerator in the calculation of variance and standard deviation.
Learn more about variance;
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Marco picked 8 1/2
pounds of apples at an orchard last weekend. He used 1/2
of the apples to make a pie for a party. Then, he used 3 1/8
pounds of apples to make applesauce. How many pounds of apples does Marco have left?
Answers
Answer: 4 [tex]\frac{7}{8}[/tex] pounds of apples
Step-by-step explanation:
We will subtract the apples used from the original pounds of apples to solve this question. First, I will turn all units into eighths to subtract.
8 1/2 -> 8 4/8
1/2 -> 4/8
3 1/8 -> 3 1/8
Next, I will subtract and simplify as I go.
8 4/8 - 4/8 - 3 1/8
8 - 3 1/8
8 - 3 = 5, 5 - 1/8 = 4 7/8
Marco has 4 7/8 pounds of apples left.
There are letter tiles in a bag. Four A’s, five B’s, six C’s and five D’s. You select one tile, replace it, and then draw another.
P( A and A)=______
A. 2/5
B. 1/25
C. 4/20
D. 1/20
Answers
Answer:
[B] 1/25
Step-by-step explanation:
From the given:
There are letter tiles in a bag. Four A’s, five B’s, six C’s and five D’s. You select one tile, replace it, and then draw another.
To Find:
P( A and A)=______
Solution:
Since it given that:
Four A’s, five B’s, six C’s and five D’s.
Thus we have:
A,A,A,A,B,B,B,B,B,C,C,C,C,C,C,D,D,D,D,D
Total number of letters tiles = 20
Probability of selecting A = 4/20 = 1/5
Probability of selecting A = 4/20 = 1/5
P( A and A)= 1/5 × 1/5 = 1/25
Thus, the answer is [B] 1/25
Kavinsky
Help me !! 10/7 x 9/8 7/9 x 8/10
Answers
Answer:
Step-by-step explanation:
10/7 x 9/8 = 90/56 = 45/28.
7/9 x 8/10 = 56/90 = 28/45
If you want to multiply these, the answer is (28*45) / 28/45) = 1.
Answer:
((((10 / 7) * 9) / 8 7/9) * 8) / 10 = 1.17179024
fraction: 7323689 / 6250000
decmal:1.17179024
+
II
alu
What is the value of x in the equation
x-2
3
1) 4
2) 6
3) 8
4) 11
Answers
[tex]\dfrac{x-2}3 +\dfrac 16 = \dfrac 56 \\\\\implies \dfrac{x-2}3 = \dfrac 56 - \dfrac 16\\\\\implies \dfrac{x-2}3= \dfrac 46\\ \\\implies \dfrac{x-2}3 = \dfrac 23\\\\\implies x -2=2\\\\\implies x =2+2 = 4\\\\\text{Hence, the value of x is 4.}[/tex]
Two sides of a parallelogram are 170 feet and 650 feet. The measure of the angle between these sides is 113°. Find the area of the parallelogram to the nearest square foot.
Answers
The area of the parallelogram is equal to 101716 ft².
QuadrilateralsThere are different types of quadrilaterals, for example: square, rectangle, rhombus, trapezoid and parallelogram. Each type is defined accordingly to its length of sides and angles. For example, the opposite sides of a parallelogram are equal and parallel.
Parallelogram AreaYou can find the parallelogram area from two formulas:
A=b * h , where b=base and h=height;A= a*b* sin(Θ), where a=side 1, b=side 2 and Θ=angle between the sides;The question gives: the length of both sides (170 ft and 650 ft) of the parallelogram and the angle between these sides (113°).
Thus, you can apply the formula: A= a*b* sin(Θ). Therefore:
A= 170 * 650* sin (113°)
A=101716 ft²
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What is 4 2/5 divided by 1 1/10
Answers
Answer:
✓ 4 2/5 divided by 1 1/10= 4
Step-by-step explanation:
✓ Factor the numerator and denominator and cancel the common factors.
✓ Hopefully this helps you!
-Keira
When [tex]4\frac{2}{5}[/tex] is divided by [tex]1 \frac{1}{10}[/tex] we get the value 4.
Convert mixed numbers to improper fractions:
[tex]4\frac{2}{5}[/tex] = (4 × 5 + 2) / 5 = 22/5
[tex]1 \frac{1}{10}[/tex] = (1 × 10 + 1) / 10 = 11/10
Divide the two fractions:
(22/5) ÷ (11/10)
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
(22/5) × (10/11)
220/55
220/55 can be reduced by dividing both the numerator and denominator by their greatest common divisor, which is 5:
(220 ÷ 5) / (55 ÷ 5)
= 44/11
=4
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A public library has an aquarium in the shape of a rectangular prism. The base is 6 feet by 2.5 feet. The height is 4 feet. How many square feet of glass were used to build the aquarium? (The top of the aquarium is open.)
Answers
Answer:
83 ft^2
Step-by-step explanation:
6*2.5 (bottom)= 15
4*2.5*2 (one set of sides)= 20
4*6*2 (another set of sides)= 48
Add, 83 ft^2
What is the radius and diameter of 13cm
Answers
Answer:
Radius = 6.5
diam = 13
Step-by-step explanation:
diameter is already stated
radius is half of the diameter
solve the following equation 6x+2+56=180
Answers
Answer:
20.3
Step-by-step explanation:
6x+2+56=180
-2 -56 -58
6x= 122
/6 /6
x=20.3
Descriptive Measures Instructions: Use the following data set to find the sample statistics for the following data set. (Round answers to the nearest hundredth, if necessary.
{51,48,42,43,48,48,46,15,29,45,47,55,46,35,47,48,54,26,53,42}
Find n
Answers
Answer:
(1) N or n = 20
(2) = 43.4
(3) σ = 9.78 and s = 10.04.
Step-by-step explanation:
Doctor Jones starts an appointment every 20 minutes.
Doctor Warholm starts an appointment every 35 minutes.
The first appointment for both doctors starts at 8.30 am.
What is the next time that they have an appointment start at the same time?
Answers
The first appointment for both doctors starts at 8:30am.
To find:Next appointment that they start at same time.
Solution:For the same time same time take LCM of 20 and 35.
20= 2x2x5
35= 5×7
hcm (20,35) = 2X2X5X7 = 140
They will take appointment after 140 minutes. 1.e. 2 hours 20 minute.
so, next appoitment will be 10:50a.m.
The next time that they have an appointment start at the same time is at 10:50 a.m.
How do we calculate the appointment time start?We are given that the first appointment for both doctors starts at 8:30am while the Next appointment also started at same time but Doctor Jones starts his every 20 minutes while Doctor Warholm starts his every 35 minutes.
First stepFor the same time, take LCM of 20 and 35.
20 = 2 * 2 * 5
35 = 5 * 7
Second stepHCM (20, 35) = 2 * 2 * 5 * 7
HCM (20, 35) = 140
Hence, it will take the appointment after 140 minutes which equals to 2 hours 20 minute.
Now, from 8.30 am, the addition of the 2 hours 20 minute gives us 10:50 a.m which is the next time that they have an appointment start at the same time.
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III
Finding a missing side length given two similar triangles
AABC and APQR are similar. Find the missing side length.
B
AA
30
48
5
8.
А
36
СР
?
R
Answers
Answer:
6
Step-by-step explanation:
These are similar triangles, which means the side lengths are proportional!
30:5=6, so there is a scale factor of six.
36/6 is six!
Answer:
PR = 6
Step-by-step explanation:
since the triangles are similar then the ratios of corresponding sides are in proportion , that is
[tex]\frac{AC}{PR}[/tex] = [tex]\frac{BC}{QR}[/tex] ( substitute values )
[tex]\frac{36}{PR}[/tex] = [tex]\frac{48}{8}[/tex] = 6 ( multiply both sides by PR )
6 PR = 36 ( divide both sides by 6 )
PR = 6
Can someone please help me? :(
Answers
Answer:
A
Step-by-step explanation:
4ft*4 1/3ft*3 1/3ft = 57 7/9 ft3
Which graph would be produced by k*f(x) if f(x)=x squared k
x= -1/2
Answers
Answer:The square below represents one whole.
What percent is represented by the shaded area?
Step-by-step explanation:
Factor 5x² + 8x – 4 using substitution.
Answers
Answer:
(
5
x
−
2
)
(
x
+
2
)
is the factorised form of the expression.
Explanation:
5
x
2
+
8
x
−
4
We can Split the Middle Term of this expression to factorise it.
In this technique, if we have to factorise an expression like
a
x
2
+
b
x
+
c
, we need to think of 2 numbers such that:
N
1
⋅
N
2
=
a
⋅
c
=
5
⋅
(
−
4
)
=
−
20
AND
N
1
+
N
2
=
b
=
8
After trying out a few numbers we get
N
1
=
10
and
N
2
=
−
2
10
⋅
(
−
2
)
=
−
20
, and
10
+
(
−
2
)
=
8
5
x
2
+
8
x
−
4
=
5
x
2
+
10
x
−
2
x
−
4
=
5
x
(
x
+
2
)
−
2
(
x
+
2
)
(
x
+
2
)
is a common factor to each of the terms
=
(
5
x
−
2
)
(
x
+
2
)
(
5
x
−
2
)
(
x
+
2
)
is the factorised form of the expression.
What is the measure of each angle in the circle?
40°
60°
45°
36°
Answers
Answer:
i think it would be 45
hope im right
if not i can change it
Step-by-step explanation:
Answer:
45°
Step-by-step explanation:
The circle is divided into 8 equal arcs, each arc measuring an angle of:
360/8 = 45°
Hope this helps