Answers
The horizontal axis is the real axis and the vertical axis is the imaginary axis. We find the real and complex components in terms of r
and θ
where r
is the length of the vector and θ
is the angle made with the real axis.
From Pythagorean Theorem :
r2=a2+b2
By using the basic trigonometric ratios :
cosθ=ar
and sinθ=br
.
Multiplying each side by r
:
rcosθ=a and rsinθ=b
The rectangular form of a complex number is given by
z=a+bi
.
Substitute the values of a
and b
.
z=a+bi =rcosθ+(rsinθ)i =r(cosθ+isinθ)
In the case of a complex number, r
represents the absolute value or modulus and the angle θ
is called the argument of the complex number.
This can be summarized as follows:
The polar form of a complex number z=a+bi
is z=r(cosθ+isinθ)
, where r=|z|=a2+b2−−−−−−√
, a=rcosθ and b=rsinθ
, and θ=tan−1(ba)
for a>0
and θ=tan−1(ba)+π
or θ=tan−1(ba)+180°
for a<0
.
Example:
Express the complex number in polar form.
5+2i
The polar form of a complex number z=a+bi
is z=r(cosθ+isinθ)
.
So, first find the absolute value of r
.
r=|z|=a2+b2−−−−−−√ =52+22−−−−−−√ =25+4−−−−−√ =29−−√ ≈5.39
Now find the argument θ
.
Since a>0
, use the formula θ=tan−1(ba)
.
θ=tan−1(25) ≈0.38
Note that here θ
is measured in radians.
Therefore, the polar form of 5+2i
is about 5.39(cos(0.38)+isin(0.38))
.
Related Questions
The powers of I discussion. see photo for question!!
Answers
Answer:
it is, in fact, i
Step-by-step explanation:
first of all, let's remember that
[tex]i^{0} = 1 \\ {i}^1 = i \\ {i}^{2} = - 1 \\ {i}^{3} = - i[/tex]
and follow up powers follow the same pattern. at this point let's look at the exponent: you should know that if a number is a multiple of 4, his last two digits are a multiple of 4. in our case,
[tex]5689 = 5688 + 1 = 4k + 1[/tex]
and honesly, we do not care about what k is. rest is playing with power rules
[tex] {i}^{5689} = {i}^{5688 + 1} = {i }^{4k + 1} = { ({i}^{4}) }^{k} \times i = i[/tex]
Solve for x. 3 - 4x = 18 + x
Answers
Answer:
x= -3
Step-by-step explanation:
subtract the x and subtract it to -4x
3-5x=18
subtract the three and subtract three from 18
-5x=15
divided the -5 on both sides
x=-3
Which expression is equivalent to 16
O -(16)4
O 164
√ F
16
(76) ?
-(+)*
16
Answers
The expression that is equivalent to (1/16)^-4 is 16^4. Option B
What are index forms?Index form of a number is mathematically described as the number that is written as an exponential expression or that is raised to a variable or another number.
The exponent of an expression explains the multiple times in which the base replicates itself to evaluate the expression.
It is also called the scientific notation or standard form of a number.
From the information given, we have that;
(1/16)^-4
Using the indices rule;
(a^-n) is same as (1/a)^n
So, if we have (1/16)^-4, it is equivalent to;
16^4
Hence, the value is 16^4
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Complete question:
Which expression is equivalent to (1/16)^-4?
The answer choices are
A) (-16)^4
B) 16^4
C) 4 square root 1/16
D) -(1/16)^-4
What do I put as the response?
Answers
Step-by-step explanation:
the law of intersecting lines : the angles between both lines are the same on both sides of either line, just left-right mirrored.
that's why the angles are while going around the intersection point (starting with the known 37° angle) :
37° - supplementary angle - 37° - supplementary angle
and again from the start.
"supplementary" angles are pairs of angles that together are 180°. just like 2 neighboring angles at intersecting lines.
so, 37° = 7x - 5
example 4
116° is the supplementary angle of 4x.
that means
180 = 116 + 4x
64 = 4x
x = 64/4 = 16
If the money obtained for 70 empty bottles can buy 35 pencils .How many pencils would the money from 90 empty bottles buy?
Answers
Since 70/35= 2
90/2 = 45
answer the question correctly for brainliest
Answers
Answer:
15p + 10
Step-by-step explanation:
5 (3p+2)
15p + 10
(Look at picture) Ty for the help
Answers
x = 3y perendicular to l
y = -3x parallel to l
y = 3x + 2 neither
What is Slope ?A line's slope, often known as its gradient, is a numerical representation of the line's steepness and direction.
The increase divided by the run, or the ratio of the rise to the run, is known as the line's slope. In the coordinate plane, it describes the slope of the line. Finding the slope between two separate locations and calculating the slope of a line are related tasks.
One approach for determining a line's slope from a graph is to use the formula immediately after knowing the coordinates of two points that are on the line. Let's assume that the coordinate values for the two spots are unknown. Therefore, we also have another way to determine the slope of the line. This approach entails attempting to determine the tangent of the angle formed by the line and the x-axis.
The line is y = -3x -1
slope is -3 by slope intercept form
The line x = 3y
or y = (1/3)*x
slope is 1/3
we know for a line perpendicular to a given line slope should be -1/m so the slope will be (1/3)
so, x = 3y is perpendicular
y = -3x have slope -3 so it is parallel
y = 3x + 2 slope is 3 so neither
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In a right triangle, cos (9x) = sin (6x - 10)°. Find the larger of the triangle's two
acute angles.
Answers
The larger acute angle in the right triangle is 60°
What is a right triangle?A triangle with at least one angle a right angle, is called a right triangle.
Given that, In a right triangle, cos (9x) = sin (6x - 10)°.
We know that, sine and cosine of an angle are supplementary.
Therefore,
9x+6x-10 = 90
15x = 100
x = 20 / 3
Therefore, sin(6(20/3)-10)°
= sin(30)°
And also,
cos(9(20/3)
= cos(60)°
Hence, the larger angle is in the triangle is 60°
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A City Park is in the Shape of a triangle where the sides have Lengths 163m, 145m, and 145m.if a fence was built around the perimeter the park, how long would the fence need to be?
Answers
The required fence to be built around the perimeter of the park should be 453 m long.
What is the triangle?The triangle is a geometric shape that includes 3 sides and the sum of the interior angle should not be greater than 180°.
Here,
A City Park is in the Shape of a triangle where the sides have Lengths of 163 m, 145m, and 145m.
The perimeter of the triangle = sum of all sides
= 163 + 145 + 145
= 453 m
Thus, the required fence to be built around the perimeter of the park should be 453 m long.
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A spherical water tank has a radius of 25 feet. Can it hold enough water to fill a swimming pool 50 meters long, 25 meters wide with a constant depth of 2 meters?
Show your work by comparing each cubic feet
Answers
The spherical water tank can not hold enough water to fill up the swimming pool, given the length, width, and constant depth of the water.
How to find the amount of water ?The amount of water that the spherical tank can hold is the volume of the spherical tank. This volume can be found by the formula:
= 4 / 3 x Pi x radius ³
The volume of this spherical tank is therefore :
= 4 / 3 x Pi x 25 feet
= 65, 450 feet ³
Then, to see if it can hold enough water for a swimming pool 50 meters long, 25 meters wide with a constant depth of 2 meters, find the volume of the swimming pool as:
= Length x Width x Depth
= 50 x 25 x 2
= 2, 500 meters ³
1 m³ to feet ³ is 35.3147 so the volume in feet is:
= 2, 500 x 35.3147
= 88, 287 feet ³
The water tank can not hold enough water for the swimming pool.
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In the schoolyard we were 80 children playing soccer. After a while more children signed up and now we are 22 children in the camp. How many new children have come?
Answers
Answer: 21 new children have come.
Step by step explanation:Data :
In the schoolyard we were = 80 children playing soccerAfter a while more children signed up = and there were 22 childrenHow many new children have come = ?We simply have to subtract the number of children playing from the number of children who arrived after a while:
[tex]\bf \: 43 - 22 = 21[/tex]
We check:
[tex]\bf \: 22 + 21 = 43[/tex]
Rpt: 21 new children have come.
What is the arithmetic meaning of set of data 6,1,5,8and 10
Answers
The arithmetic mean of the set of data 6, 1, 5, 8 and 10 is calculated to be 6
What is arithmetic mean?The simplest and most popular way to measure a mean or average is with the arithmetic mean. It simply entails adding up a collection of numbers, then dividing that total by the total number of numbers in the series.
Adding up the numbers
= 6 + 1 + 5 + 8 + 10
= 30
The total number of items in the data is = 5
The arithmetic mean calculation
arithmetic mean = sum of numbers / number of items
arithmetic mean = 30 / 5
arithmetic mean = 6
The arithmetic mean is 6
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Assume A varies directly as B^2. If B is multiplied by 3.55, how will it affect A. In other words: if we multiply B by 3.55, by how much would we have to multiply A, in order to preserve the same coefficient of proportion?
Write your answer as a decimal rounded to two decimal places.
Answers
Answer: 12.60
Step-by-step explanation: A = B^2. If B is multiplied by 3.55, then A = (3.55B)^2. That gives us A = 12.6025B. Round 12.6025 to hundredth and you get 12.60
Melissa has a student dictionary on her jasper dictionary connect 76 pages and the dictionary more word start with the letter S. then any other letter I'll keep on 13 full pages she opens up the dictionary random what is the proper ability that the page can contain words starting with the word s
Answers
In the figure shown, t || x and k || w and m<3 = 20°. Choose the list which includes all of the other angles that measure 20 degrees. ܔܔ
O A 1
OB. 1,5,7
O c. 1,14, 16
O D. 1,5,7, 9, 11, 14, 16
Answers
The angles which measure 20 degrees are∠1 ,∠5 ,∠7 ,∠9 ,∠11 ,∠14 ,∠16. The solution has been obtained using the properties of parallel lines.
What are parallel lines?Parallel lines are any two or more lines that lie in the same plane, are equally spaced apart, and never cross one another.
We are given t || x and k || w and m∠3 = 20°
Using the properties of parallel lines, we get -
∠3 = ∠1 (As these are vertically opposite angles)
∠3 = ∠5 (As these are alternate interior angles)
∠3 = ∠7 (As these are corresponding angles )
Now,
∠4 = 180 - ∠3 (As they form a linear pair)
∠10 = 180 - ∠9 (As they form a linear pair)
Also,
∠10 = ∠4 (Parallelogram's opposite angles are same)
So,
180 - ∠9 = 180 - ∠3
∠3 = ∠9 .... [ii]
∠9 = ∠11 (As these are vertically opposite angles)
∠3 = ∠11 ( from [ii] ) .... [iii]
∠11 = ∠14 (As these are alternate interior angles)
∠3 = ∠14 ( from [iii] ) .... [iv]
∠14 = ∠16 (As these are vertically opposite angles)
∠3 = ∠16 ( from [iv] )
Hence, the angles which measure 20 degrees are
∠1 ,∠5 ,∠7 ,∠9 ,∠11 ,∠14 ,∠16.
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. 10 people can fit comfortably into a 7 x 12 feet square. What is the ratio of number of people and the
rectangle's area?
Answers
Answer:
5 people per 42 square feet.
Step-by-step explanation:
The area of the rectangle is 7 x 12 = 84 square feet.
The ratio of the number of people to the area of the rectangle is:
10 people : 84 square feet
You can simplify this ratio by dividing both sides by 2:
10/2 people : 84/2 square feet
Which can be written as
5 people : 42 square feet
Answer:
The ratio of the number of people to the rectangle's area is 10 people : 84 square feet.
Step-by-step explanation:
The ratio of the number of people to the rectangle's area is 10 people : 84 square feet.
It is important to note that this ratio represents the maximum number of people that can fit comfortably in the 7 x 12 feet square. The actual number of people that can fit in the space will depend on various factors such as the size and layout of the room, the presence of furniture, and the size of the people.
For each expression, combine
like terms and write an equivalent
with fewer terms.
expression
a. 4x + 3x
b. 3x + 5x - 1
c. 5 + 2x + 7 + 4x
d. 4 - 2x + 5x
e. 10x 5 + 3x - 2
-
Be sure to identify the problem to which each
answer belongs.
Answers
Answer:
a) 7x
b) 8x-1
c) 6x=12
d) 3x+4
e) 13x-7
Step-by-step explanation:
To collect the like terms, you combine similar terms. e.g- 4x+3x would be 7x because there are 7 x's.
Suppose that you are planning a winter vacation. You want to spend a lot of time outdoors, so you are checking the "average" temperature in different locations. Would you prefer to have the "average" temperature reported as a mean, a median or a mode? Why?
Answers
The statistical measure which would best represent the average temperature as required is; Mean.
Which measure of central tendency best represents the average temperature?It follows from the task content that the measure of central tendency which best represents the average temperature is to be determined.
On this note, although the mean and median are statistical measures of the central tendency, the mean is a more accurate measure of the average temperature as in this context.
The mode on the other hand, is a statistical measure of spread.
Ultimately, while checking the average temperature in different locations, it follows that the average temperatures in each of the locations is reported as a mean.
This follows from the fact that the mean is a representative of all temperatures recorded in any one location.
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Consider the equation below. − 5 n + 31 = − 14 n − 5 Select the equation that has the same solution. − 1 − 22 n = − 20 n − 9 3 − 6 n + 3 n = − 3 + 4 − 4 n 1 = 0.75 n + 3.25
Answers
The equation which having the solution n = -4 is,
11/4n + 2 = -3 + 3/2n.
What is solution of an equation?
Any value of the variable that satisfies the equality, that is, makes the Left Hand Side (LHS) and the Right Hand Side (RHS) of the equation equal, is a solution of the equation. Finding an equation's solution or solutions is known as solving the equation.
Consider, the given equation -5n + 31 = -14n -5
Solving this equation,
14n - 5n = -5 - 31
9n = -36
n = -4
We have to find the equation which having the solution n = -4.
Consider, the first equation
-1 - 22n = -20n - 9
Solving this, we get
n = 4
Consider, the second equation
3 - 6n + 3n = -3 + 4 - 4n
Solving this, we get
n = -2
Consider, the third equation
1 = 0.75n + 3.25
Solving this, we get
n = 3
Consider, the fourth equation
11/4n + 2 = -3 + 3/2n
Solving this, we get
n = -4
Hence, the equation which having the solution n = -4 is,
11/4n + 2 = -3 + 3/2n.
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Identify the domain of f.
For which values a does lim x -> a f(x) not exist? Find all values a where lim x -> a f(x) exists, f is defined at a, but lim x -> a f(x) does not equal f(a).
Answers
The graph shows that the domain and range of the line segment were [-5, 1] & [-4, 7, correspondingly.
Describe domain.A function's domain is the set of values that it can accept. These figures represent a function like f's x-values (x). The domain of a function is the set of possible values on which it can operate. This set is the value which the return type after the x value has been entered. Y = f is a function where x is an independent variable and y is a dependent variable (x). If a value of x can be successfully utilized to produce a single value of y using a value of f, then that value is said to be within the domain of that function.
Here,
The domain denotes all conceivable possibilities for x, whereas the range denotes all possible values of y.
The graph shows that the domain and range of the line segment is [-5, 1] & [-4, 7, respectively.
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Which option is most similar in relation to the following word pair water:ocean
Answers
Water and ocean are two closely related words, as the ocean is the largest source of water on the planet.
Which option is most similar in relation to the following word pair water:ocean?Water is essential to life, and the ocean is the planet's largest reservoir of water.It covers over 70 percent of Earth's surface and contains 97 percent of the world's water. Water and ocean are linked in many ways.The ocean produces more than half of the oxygen in the atmosphere, and the ocean's currents and tides help regulate the climate.Water from the ocean evaporates into the atmosphere, eventually returning as rain, snow, and hail and replenishing the planet's fresh water supply.The ocean is home to a variety of organisms and species that rely on the water for their survival, as well as other resources such as food and shelter.Additionally, the ocean is an important source for human activities such as fishing, transportation, and recreation.In short, water and ocean are inextricably linked, and their relationship is essential to the health of the planet.To learn more about word pair water:ocean refer to:
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Enter the signs by using the corresponding letters in the correct order. Can you make this a true equation?
Answers
The corresponding letters based on the signs in their correct order that makes the numbers and operators a true equation is as follows;
((12 B E 16) C 17) D 62 F 1/2
What is the order of operations?The order of operation which indicates the proper sequence of the operations on the numbers in the equation, (PEMDAS), is as follows;
P; Parenthesis first(), then E; Exponents, followed by M or D, Multiplication or Division (Left to Right), and A or S; Addition or Subtraction.
The specified numbers are; 12, 16, 17, 62, and 1/2
The operators are; +, ×, √, -, ÷, and =
The number that is a perfect square is 16 = 4²
Applying the square root to 16, and multiplying the result to 12, we get;
12 × √(16) = 48
Subtracting 17 from the above result, we get;
48 - 17 = 31
Dividing the above result of 31 by 62, we get;
31/62 = 1/2
The operations on the numbers that make the set of numbers and operators an equation are therefore;
(12 × √(16)) - 17) ÷ 62 = 1/2
The corresponding letters from the set of letters and operators, that can make this a true equation is therefore;
B = '×', E = '√', C = '-', D = '÷', F = '='
12 B E 16 C 17 D 61 F 1/2
Applying the bracket symbol that indicates the order of operations, we get;
((12 B E 16) C 17) D 61 F 1/2
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what is ⊕ this symbol
Answers
Answer: The symbol ⊕ is known as the XOR (Exclusive Or) operator.
Step-by-step explanation:
The symbol ⊕ is known as the XOR (Exclusive Or) operator. It is a logical operator that compares two binary digits or bits and returns a 1 if one of the bits is 1 but not both. It returns 0 if both bits are the same (either 0 or 1). It is commonly used in digital logic and computer science, particularly in the field of cryptography. In simple terms, it compares two bits and returns 1 if they are different and 0 if they are the same.
Pls help Darnell is an election officer. On election day, he travels to the polling place, which is 4.4 miles away from his home. On a map of Harrison County, these two places are 8 inches apart. What is the scale of the map? Write your answer in simplest form using whole numbers.
Answers
Answer:
The scale of the map is 1 : 34848---------------------------------
Real distance is 4.4 miles and on the map it is 8 inches.
The scale is:
8 in : 4.4 miles = Divide both sides by 81 in : 0.55 milesor
1 in : 0.55 * 63360 in = Convert 1 mile = 63360 inches1 : 34848(x - 12)(x + 5)/(x - 3)³≤0
Answers
The range of values of x for the given expression (x - 12)(x + 5)/(x - 3)³≤0 is ,
( -∞ , -15] and [3, 12]
What are inequalities?Inequalities are the comparison of mathematical expressions, whether one quantity is greater or smaller in comparison to another quantity.
We use these symbols to represent inequalities, '>' , '<', '≥', '≤'
Given expression,
(x - 12)(x + 5)/(x - 3)³ ≤0
by using wavy curve method it can be state that,
function hold negative value,
1. when x is less than -15
⇒ x ≤15
2. when x is in between
⇒ 3≤ x ≤12
Hence, range of x is ( -∞ , -15] and [3, 12]
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Two girls divided $1.60 in the ratio 5 : 3. How much more does one girl get than the other?
Answers
let's convert those $1.60 to pennies, that's 160 pennies, now, let's divide those 160 by (5 + 3) and distribute between the girls accordingly
[tex]\stackrel{Girl1}{5}~~ : ~~\stackrel{Girl2}{3} ~~ \implies ~~ \stackrel{Girl1}{5\cdot \frac{160}{5+3}}~~ : ~~\stackrel{Girl2}{3\cdot \frac{160}{5+3}} ~~ \implies ~~ \stackrel{Girl1}{5\cdot 20}~~ : ~~\stackrel{Girl2}{3\cdot 20} \\\\\\ \stackrel{Girl1}{100}~~ : ~~\stackrel{Girl2}{60}\qquad \textit{one girl got \underline{40 more cents } than the other girl}[/tex]
The results of a survey of students showed that 53 students had a Netflix account, 46 students had a Hulu account, and 87 students has a Netflix or a Hulu account. How many students had both a Netflix and Hulu account? Show how you got your answer please!
Answers
The students that have both a Netflix and Hulu account is 12.
How to calculate the number?We can use the principle of inclusion-exclusion to solve this problem. The principle states that if we want to know how many students have either a Netflix or a Hulu account (or both), we can add the number of students who have a Netflix account to the number of students who have a Hulu account and then subtract the number of students who have both.
So, using the information given:
Netflix or Hulu = Netflix + Hulu - (Netflix and Hulu)
87 = 53 + 46 - (Netflix and Hulu)
To find the number of students who have both a Netflix and Hulu account we can solve for (Netflix and Hulu) :
(Netflix and Hulu) = Netflix + Hulu - 87
(Netflix and Hulu) = 53 + 46 - 87
(Netflix and Hulu) = 12
So, 12 students have both a Netflix and Hulu account.
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A child uses her hand to measure the width of a tabletop. Her hand has a width of 8.3 cm at its widest point, and she finds the tabletop to be 15.5 hands wide.
Answers
The width of the table top is 132 cm and 1.32 in metres.
What is width?The shorter distance of an object or a figure is its width, which indicates how broad or wide the given figure is. Additionally, width is measured in linear units such as metres, centimetres, inches, and so forth.
Length: Length is used to calculate the separation between two points. The longest dimension of a figure, length, indicates how long an object or figure is. It is measured in linear units like metres, centimetres, inches, and so forth.
Width of hand = 8cm
No of hands = 16.5 hands
Width of tabletop = Width of hand * No. of hands = 8cm × 16.5 = 132cm = 1.32m
The width of the table top is 132 cm and 1.32 in metres
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ayudaaa a resolver esto plis
Answers
The value of x in degrees will be 6° and 11° using properties of parallel lines.
What Qualities Characterize Parallel Lines?They are typically equal-distance, parallel straight lines. The lines here are parallel. They never come together no matter how far you extend them in any one way.
Pairs of equal-sized angles that are both obtuse or acute and are generated on the same side of the transversal are known as corresponding angles. Each pair of comparable angles on the same side of the crossing transversal is equal to the other.
The alternative angle theorem states that alternate interior angles or alternate exterior angles that come from cutting two parallel lines are congruent.
In the above figure,
8x + 7 = 55
8x = 55 - 7
x = 6
Measure of the angle will be 8 × 6 + 7 = 49
In 11)
10x + 10 = 9x + 21 (by using first corresponding angles and then alternate angles)
⇒x = 11
The value of x in degrees will be = 11° using properties of parallel lines.
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On a team, 7 girls and 6 boys scored a total of 79 points. The difference between the number of points scored by the 7 girls and the number of points scored by the 6 boys is 19. Each girl scored the
same number of points and each boy scored the same number of points. Find the number of points scored by each girl and each boy.
Answers
Each girl scored 9 points
Each boy scored 7 points
Step-by-step explanation:
Let the number of points each girl scored = a
Let the number of points each boy scored = b
7 girls and 6 boys scored a total of 105 points.
7a + 6b = 105......... Equation 1
The difference between the number of points scored by the 7 girls and the number of points scored by the 6 boys is 21
7a - 6b = 21........... Equation 2
7a + 6b = 105......... Equation 1
7a - 6b = 21........... Equation 2
Subtract Equation 2 from Equation 1
12b = 84
b = 84/12
b = 7
Therefore, each boy scored 7 points
7a + 6b = 105......... Equation 1
b = 7
7a+ 6b = 105
7a + 6(7) = 105
7a + 42 = 105
7a = 105 - 42
7a = 63
a = 63/7
a = 9
Therefore, each girl scored 9 points.
