To derive the trigonometric Fourier series from the complex exponential series, we can start with the complex exponential Fourier series: The trigonometric Fourier series is f(x) = a0/2 + Σ[cn e^(inx)]
where cn = (an - ibn)/2.
f(x) = a0/2 + Σ(an cos(nx) + bn sin(nx))
where a0/2 is the average value of f(x), and an and bn are the Fourier coefficients given by:
an = (1/π) ∫f(x)cos(nx)dx
bn = (1/π) ∫f(x)sin(nx)dx
We can rewrite the trigonometric terms in terms of complex exponentials as follows:
cos(nx) = (e^(inx) + e^(-inx))/2
sin(nx) = (e^(inx) - e^(-inx))/(2i)
Substituting these expressions into the complex exponential Fourier series, we get:
f(x) = a0/2 + Σ[(an + ibn)(e^(inx) + e^(-inx))/2]
where ibn = bn/i.
We can simplify this expression as follows:
f(x) = a0/2 + Σ[cn e^(inx)]
where cn = (an - ibn)/2.
This is the trigonometric Fourier series, which expresses the function f(x) as a sum of complex exponential terms with real coefficients. We can write this more explicitly as:
f(x) = a0/2 + Σ[cn (cos(nx) + i sin(nx))]
which is the same as:
f(x) = a0/2 + Σ[cn cos(nx)] + i Σ[cn sin(nx)]
So, to derive the trigonometric Fourier series from the complex exponential series, we simply substitute the complex exponential expressions for cos(nx) and sin(nx), and simplify the resulting expression to obtain the coefficients cn.
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PLEASE HELP!! The Great Pyramid of Cheops has a square base whose sides each measure 230 meters. The altitudes of the triangular faces of the pyramid each make an angle of 51.8 degrees with the base.
What is the height of the pyramid? What is the shortest distance you would have to climb up a triangular face to reach the top?
Triangular (comparative more triangular, superlative most triangular) (comparative more triangular, superlative most triangular) a triangle-shaped form.
What is the height of the square pyramid?The entire area bounded by any triangle's three sides is referred to as the triangle's area. In essence, it is equivalent to 1/2 of base times height, or A = 1/2 b times h.
The Great Pyramid held the record for the world's highest building for more than 4,000 years, standing at 146.5 m (481 ft) tall. After losing 9.5 meters (31 feet) off the top, it is now 137 meters (449.5 feet) high.
S2(A2)2 is a formula that calculates the height of a square pyramid. In this example, a represents the length of the square base edge and s represents the square pyramid's slant height. Given by a2(h3) is the volume of the square pyramid.
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You need to strengthen a 22% alchohol solution with a pure alcohol solution to obtain a 40% solution. How much pure alcohol should you add to 100 milliliters of the 22% solution?
To obtain a 40% alcohol solution from a 22% solution, you need to add 18% more alcohol.
How to find this?
100 ml of 22% alcohol solution contains 22 ml of pure alcohol and 78 ml of other ingredients.
Therefore, to obtain a 40% solution, you need 40 ml of pure alcohol in every 100 ml of solution.
So, 18 ml of pure alcohol should be added to 100 ml of 22% solution to obtain 40% solution.
Another way to calculate this is:
You can calculate the amount of pure alcohol you need to add by multiplying the initial volume of the solution (100ml) by the percentage increase in alcohol content (18%) .
100ml * 18% = 18ml
So you would need to add 18ml of pure alcohol to 100ml of 22% alcohol solution to obtain a 40% alcohol solution.
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Which triangle is a 30 degrees 60 degrees 90 degrees triangle?
Answer:
The answer is
30 degrees is acute angle
60 degrees is also acute angle
90 degrees is right angle
Step-by-step explanation:
anything below 90 degrees is acute angle
90 degrees creates a perfect square which is a right angle
If (z-1/z)=6 find the value of :
(z+1/z)
So the possible values of (z+1/z) are 6 and 0 for z=5 and z=-1 respectively.
Define Algebric expression.An algebraic expression in mathematics is an expression created using variables, constant algebraic numbers, and algebraic operations. A good example of an algebraic expression is 3x2 2xy + c.
What is variables and constants ?A constant has a fixed value and does not vary over time. For instance, the size of a shoe, a piece of clothing, or any other article of clothing will never alter. x+y = 8 is an algebraic equation, and 8 is a constant that cannot be altered. Variables are concepts that change or fluctuate throughout time.
We can start by multiplying both sides of the equation by (z+1/z) to get:
(z-1/z)(z+1/z) = 6(z+1/z)
This simplifies to:
z^2 - 1 = 6z + 6
We can then solve for z by isolating it on one side of the equation.
z^2 - 6z - 5 = 0
(z-5)(z+1) = 0
So the solutions to this equation are z = 5 and z = -1
Now we can substitute the value of z back into the original expression to find the value of (z+1/z)
For z = 5, (z+1/z) = (5+1/5) = 6
For z = -1, (z+1/z) = (-1+1/(-1)) = 0
So the possible values of (z+1/z) are 6 and 0.
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How much of the circle is shaded? Write your answer as a fraction in simplest form.
Answer:
Step-by-step explanation:
Get a common denomenator, add the two fractions together and subtract the sum from 1 whole.
What is the formula of minor sector and minor segment?
The formula of the minor sector = (θ/360°) × π r²
The formula of the minor segment = (θ / 360°) × πr² - (1/2) r² sin θ (OR) r² [πθ/360° - sinθ / 2] if 'θ' is in degrees
What is a segment of a circle?
A circle's arc and chord together define a segment, which is the area within that region. Let's go over what a circle's arc and chord signify once more.
What is a sector of a circle?
The term "Minor Sector" refers to the tiny area of a circle that is removed by two circle radii. The minor sector is a tiny area occupied by two radii.
Here,
We have to find the formula of the minor sector and minor segment.
We concluded that
The formula of the minor sector = (θ/360°) × π r²
The formula of the minor segment = (θ / 360°) × πr² - (1/2) r² sin θ (OR) r² [πθ/360° - sinθ / 2] if 'θ' is in degrees.
Hence, the formula of the minor sector is (θ/360°) × π r².
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Simplify the expression to a polynomial in standard form: (x+6)(2x^2-4x+1) (x+6)(2x 2 −4x+1)
The standard form of polynomial 2x³ + 8x² -23x +6.
What is polynomial?A polynomial is a mathematical equation that solely uses the operations addition, subtraction, multiplication, and non-negative integer exponentiation of variables. Variables are sometimes known as indeterminates in mathematics.
An expression that consists of exponents, constants, and variables that are concatenated using mathematical operations is referred to as a polynomial.
Given:
(x+6)(2x²-4x+1)
Multiplying the polynomial
= 2x³ -4x² + x + 12x² - 24x + 6
= 2x³ -4x² + 12x² + x -24x + 6
= 2x³ + 8x² -23x +6
Hence, the required polynomial is 2x³ + 8x² -23x +6.
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What is the domain for this function?
The domain of this function is all real numbers from -54 to 60.
What is real number?Real numbers are any number that can be expressed on the number line. They include natural numbers, whole numbers, integers, rational numbers and irrational numbers. Real numbers can be expressed as decimals, fractions, or in scientific notation. Real numbers can also be positive, negative, or zero. They are used in mathematics to represent a wide variety of physical quantities, such as distances, velocities, and forces.
Real numbers are numbers that represent an exact quantity, such as whole numbers and fractions. To find real numbers, you can use the standard number line or the decimal point system. The number line is a graphical representation of all real numbers, and it starts with the positive numbers on the right and the negative numbers on the left. The decimal point system assigns place values to each digit in a number, and the real numbers are the numbers that have non-repeating digits.
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Can someone please help me
Answer:
What is this on
Step-by-step explanation:
1) A survey asked 200 Grade 4 students about their reading preferences. Based on Graph #1, about how many of them would you expect prefer comic books?
about how many students prefer literature (poems, plays, or novels)?
A. 45
B. 90
C. 100
D. 130
E. 180
answer:B. 90 good luck I hope you pass
The answer is B which is 90
Find greatest common factor of 24, 60. Show your work
Answer:
12
Step-by-step explanation:
What is GCF(greatest common factor)?
The GCF of a, b is the largest positive number that divides both a and b without a remainder
Prime factors of a number are the prime numbers that when multiplied together result in the number
Prime factors of 24
24 = 2 x 12
12 = 3 x 4
4 = 2 x 2
So prime factorization of 24 are 2 x 3 x 2 x 2 = 2³ x 3
Prime factors of 60:
60 = 2 x 30
30 = 2 x 15
15 = 3 x 5
Prime factorization of 60: 2 x 2 x 3 x 5 = 2² x 3 x 5
Find the factors that are common to both 24 and 60:
2² and 3
2² x 3 = 4 x 3 = 12
So the GCF is 12
This is the greatest number that can divide 24 and 60 without a remainder
46+46 +y+ y= 180 *plug y in to solve for x
The value of y in the equation is 44.
What is equation ?
A mathematical expression with an equals sign is referred to as an equation. Algebra is widely used in equations. When performing calculations but unsure of the precise amount, algebra is used. A lot of specialists, such as air traffic controllers, architects, computer programmers, and carpenters, employ equations on a daily basis.
The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
A mathematical statement known as an equation is made up of two expressions joined together by the equal sign.
46 + 46 + y + y = 180
2y = 180 - 46 - 46
2y = 180 - 92
y = 88 / 2
y = 44
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What are the minimum and maximum values of this?
The minimum value is 0.5 and the maximum value is 3.5.
What are the minimum and maximum values of this?The minimum value of 1/n(x) + 2 over the interval 5≤ x ≤ 8 is 2.This is because 1/n(x) is a decreasing function and the smallest value it can take is 1/n(8) = 1/8. Adding 2 to this gives 2. The maximum value of 1/n(x) + 2 over the interval 5≤ x ≤ 8 is 5.This is because 1/n(x) is an increasing function and the largest value it can take is 1/n(5) = 1/5. Adding 2 to this gives 5. Overall, 1/n(x) + 2 takes on values between 2 and 5 over the interval 5≤ x ≤ 8.This is because 1/n(x) is a decreasing function over the interval and adding 2 to this makes the minimum value 2, while 1/n(x) is an increasing function over the interval and adding 2 to this makes the maximum value 5.The minimum and maximum values for the function 1n(x) +2 over the interval 5≤ x ≤ 8 can be determined by examining the graph of the function over the given interval.Since the function is increasing over the given interval, the minimum value of the function is equal to its value at the lower bound of the interval (x = 5), which is 1n(5) + 2 = 2.71.The maximum value of the function is equal to its value at the upper bound of the interval (x = 8), which is 1n(8) + 2 = 3.11.To learn more about the minimum and maximum values refer to:
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Use properties of operations to find the product of 8 and 88.
Fill in the blanks below.
8 x 88 = 8x (90-a)
= (8 x 90)-(8x a), a =
= 720-
||
A
The value of a for the given expression will be 2.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the expression will be 8 x 88 = 8x (90-a). The value of a will be calculated as,
8 x 88 = 8x (90-a)
88 = 90 - a
a = 90 - 88
a = 2
Hence, the value of a will be 2.
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Solve for x. Round to the nearest tenth, if necessary.
Select the correct answer. What is the slope of the line represented by this equation? -3x + 8y = 12 a. B. C. D.
With the help of the given equation, we know that the required slope is (A) 3/8.
What is a slope?In math, the gradient of a line is a number that tells you how steep it is and in which direction it goes in.
The point-slope form is a way of writing equations (y-y₁=m(x-x₁)) to emphasize the slope and a point on the line that isn't the y-intercept.
To highlight the same line's slope and y-intercept, you can rewrite the equation in slope-intercept form (y=mx+b).
So, we have the equation:
-3x + 8y = 12
Calculate the slope as follows:
We need to change it into a more general format.
-3x+8y=12
⇒ 8y = 3x+12
⇒ y = 3x/8+12/8
⇒ y = 3x/8+3/2
If you look at it in terms of y = mx+c, the m would be 3/8.
Therefore, with the help of the given equation, we know that the required slope is (A) 3/8.
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Complete question:
Select the correct answer. What is the slope of the line represented by this equation? -3x + 8y = 12
a. 3/8
B. 4/8
C. 8/4
D. 2
What is common midpoint method?
There are two points: one at x1, y1 and the other at x2, y2. A point with an x value of x one plus x .
what is order of rotation ?The number of rotations that a geometry can fit into is determined by its order of rotational symmetry. The degree of rotational symmetry is expressed as 360°A°, where A° is the lowest angle by which a figure may be rotated from its rotated shape to its original shape. 360 degrees The number of rotations a perfect circle can undergo while still preserving its shape is a measure of a shape's degree of rotational symmetry. A full 360° revolution won't look the same until a triangle is placed back in its initial starting position.
given
There are two points: one at x1, y1 and the other at x2, y2. A point with an x value of x one plus x .
two all divided by two lies directly between the two points. Its y coordinate is equal to y1 plus y2 divided by 2.
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Help pls and ty
What’s the slope of 6x+8y=10
What’s the y-intercept
Answer:
slope=-3/4
y=5/4
Step-by-step explanation:
attached wrk here
What is the additive inverse of 6 /- 13?
The additive inverse of 6 /- 13 is -6 /+ 13, which means that the two numbers have the same magnitude but opposite signs.
The additive inverse of a number is the number that needs to be added to the original number to get a result of zero. In other words, the additive inverse of a number is the opposite of the number in terms of sign. To find the additive inverse of 6 /- 13, we must multiply the negative sign of -13 by -1 to change it to a positive. This means that the additive inverse of 6 /- 13 is -6 /+ 13. This means that the two numbers have the same magnitude but opposite signs. To verify this, we can add the two numbers together. 6 /- 13 + -6 /+ 13 = 0. Therefore, the additive inverse of 6 /- 13 is -6 /+ 13.
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There is a soccer tournament with 16 teams. Each team must play every other team. You want to know how many games will be played. Explain how you would create a simpler problem to help solve this problem.
it’s 12
Step-by-step explanation:
Answer:ontinuing this process shows that for 16 teams, there are 8 + 4 + 2 + 1, or 15 total games. Using brackets can be done for any number of teams.
Step-by-step explanation:
in what form, typically, is the 2nd derivative of a multivariable function?
The second derivative test for a multivariable function is generally in a positive definite quadratic form.
What is the second derivative of a multivariable function?In mathematics, the second partial derivative test is a method in multivariable calculus which used to determine if a critical point of a function is a local minimum, maximum or saddle point.
A partial derivative of a multivariable function is a derivative with regard to one variable with other variables adhered constant. It is used to describe the concepts of gradient, divergence, and curl in terms of partial derivatives.
The second derivative of a function is directly the derivative of the function's derivative. Let's consider, as an example, the function f(x) = x^3 + 2x^2.
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What is the height of a triangle if its base is 24 cm and its area is 216 cm2?
The height of a triangle is calculated to be 18 cm if its base is 24 cm and its area is 216 cm².
Given that,
Base = 24 cm
Area = 216 cm²
Height = ?
The formula to find out the area of the triangle is given by,
Area A = 1/2 × Base b × Height h
A = 1/2 × b × h
Making h as subject, we have,
h = (2 A)/b = (2× 216)/24 = 18 cm
Thus, the required height of a triangle is calculated to be 18 cm if its base is 24 cm and its area is 216 cm².
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7points more than twice the number of points scored by the other team
The given statement as an expression is y = 7 + 2x
How to represent the statement as an expressionFrom the question, we have the following parameters that can be used in our computation:
A mathematical statement
The mathematical statement is given as
7 points more than twice the number of points scored by the other team
Represent the points with x and y
So, we have the following representation
7 points more than twice x is y
is equal to implies =
So, we have:
7 points more than twice x = y
more than implies +, twice implies 2 times
This gives
7 + 2 * x = y
Evaluate
7 + 2x = y
Rewrite as
y = 7 + 2x
The question does not require that we solve the expression
So, we leave it at y = 7 + 2x
Hence, the required expression is y = 7 + 2x
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if there are 20 oranges in one basket then how man in 12 baskets use the distribuive strategy to solve
Answer: There would be 240 oranges in 12 baskets.
Step-by-step explanation:
So, if there's 20 oranges per basket, you would just need to multiply 20 oranges by the 12 baskets.
20 x 12 = 240.
Convert this equation into standard form
A conversion of this equation in point-slope form into standard form is y = x/2 + 1.
What is the slope-intercept form?In Mathematics, the slope-intercept form of a line can be calculated by using this linear equation:
y = mx + c
Where:
x and y represents the data points.m represents the slope of a line.c represents the y-intercept of a line.Based on the information provided about the line, it is represented by this equation;
y - 3 = -1/2(x + 4)
Making y the subject of formula in the above equation, we have the following:
y = x/2 - 2 + 3
y = x/2 + 1
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ASAP.... Please Help I will give BRAINLIEST
[tex]{ \qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
The angle measures of the triangle are :
45°, 45° and 90° respectivelyAnd since it has an angle measuring 90° (right angle), so conclusion can be made that the given triangle is a right angled triangle.
Hence, correct choice is ;
[tex]\qquad \sf \dashrightarrow \: \bf right[/tex]
13) AMNP-ARST
a) Find the value of x if MN = 5x + 10, MP-5, RT-4, RS= 6x-8, and ST-42.
b) What is the length of MN, RS, and NP?
MN =
RS=
= 80
x=
NP =
what is number 13
MN, RS, and NP in this triangle are, respectively, 410, 400, and 320 when MN = 5x + 10.
what is triangle ?Given that it has three sides and three vertices, a triangle is a polygon. It is one of the fundamental geometrical forms. Triangle ABC is the term used to describe a triangle containing the vertices A, B, and C.A triangle is a polygon because it has three sides and three corners. The points at which the three sides of the triangle converge are referred to as its corners. Three triangle angles can be multiplied to get 180 degrees.
given
as AMNP-ARST
the x = 80
so MN = 5 * 80 + 10
= 410
RS = 6* 80 - 8
= 400
MN, RS, and NP in this triangle are, respectively, 410, 400, and 320 when MN = 5x + 10.
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PLEASE HELP!!
A hobby club runs remote-controlled boats in a tank that is shaped as shown. What volume of water can
this tank hold, to the nearest cubic metre?
The total volume of the tank is 400π + 78.72 cubic metres. To the nearest cubic meter, it is approximately 400π + 79 cubic metres.
What is the volume of a cylinder?
The volume of a cylinder is the amount of space inside the cylinder. It can be calculated by multiplying the area of the base by the height of the cylinder. The formula for the volume of a cylinder is V = πr²h, where r is the radius of the base of the cylinder and h is the height.
The tank is shown as a cylinder with a rectangular prism on top. To find the volume of water the tank can hold, we will need to find the volume of the cylinder and the rectangular prism separately, and then add them together.
The cylinder has a radius of 5 metres and a height of 8 metres. The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height. So the volume of the cylinder is V = π(5²)(8) = 400π cubic metres.
The rectangular prism has a length of 4.2 metres, a width of 2.2 metres and a height of 8 metres. The volume of a rectangular prism is given by the formula V = lwh, where l is the length, w is the width and h is the height. So the volume of the rectangular prism is V = (4.2)(2.2)(8) = 78.72 cubic metres.
So the total volume of the tank is 400π + 78.72 cubic metres. To the nearest cubic meter, it is approximately 400π + 79 cubic metres.
Please note that since π is an irrational number, the value of 400π is not an exact value, but an approximation.
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Evaluate for x = 2. 12x+84
Answer:
108
Step-by-step explanation:
x=2
12x+84
12(2)+84
24+84=108
Jamal is planting seeds for a garden nursery. He plants 9 seeds in each container. If jamal has 296 seeds to plant about how many containers will he use? Please tell me the answer I need help.
Answer:
33 containers.
Step-by-step explanation:
Dividing 296 by 9 gives you 32.89, rounding up to the next whole number gives you 33 containers.