Function Mathematics

The expression that is equivalent to (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) ) is option D. (1 + cos(x))(sin (x))

The expression (1 + cos(x))2Tangent (StartFraction x Over 2 EndFraction) is equivalent to (1 + cos(x))(sin (x)) because of the double angle identity for tangent.

The double angle identity states that tangent of 2 times an angle is equal to 2 times the tangent of that angle divided by 1 minus the square of the tangent of that angle. In other words,

tan(2θ) = 2tan(θ)/(1 - tan2(θ))

In this expression, we have tangent of x/2, so substituting θ = x/2 gives us:

tan(x) = 2tan(x/2)/(1 - tan2(x/2))

Since cos(x) = 1 - 2sin2(x/2), we can simplify the expression to:

(1 + cos(x))2tan(x/2) = (1 + 1 - 2sin2(x/2))2tan(x/2) = (2 - 2sin2(x/2))(2sin(x/2)/(1 - sin2(x/2)))

Expanding the product of the two factors gives us the final result:

(1 + cos(x))2tan(x/2) = (2 - 2sin2(x/2))(sin(x)) = (1 + cos(x))(sin(x))

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Answer:

Joe mum so gaeee

Step-by-step explanation:

your answer is 1000

Answer: To calculate the amount of pasta needed for eight servings, we need to multiply the original amount of pasta in the recipe by 8/6.

So, 1.75 cups * 8/6 = 2.3 cups of pasta.

Therefore, Rosa will use 2.3 cups or four and one quarter cups of pasta if she wants to make eight servings.

Step-by-step explanation:

Let's consider what the question asks for:

 --> perimeter of a rectangle

To find the perimeter of the rectangle:

 --> we need to know the side length

Let's consider how to find our side length:

 --> in a rectangle

     --> opposite parallel sides are equal in length

           --> in a mathematical equation, we get:

                  [tex]4x-4y=3x+5y\\2-2y=x-3y[/tex]

Now we notice,

  --> in the second equation, there is only one 'x'

        --> therefore if we find a y-value to substitute into the 'x'

            --> we can solve

Let's use the first equation to see how 'x' and 'y' relate to each other:

  [tex]4x-4y=3x+5y\\4x-3x=4y+5y\\x=9y[/tex]

Let's use that x-value and plug it the second equation:

  [tex]2-2y=x-3y\\2-2y=(9y)-3y\\2-2y=6y\\2=8y\\\\y=\dfrac{1}{4}[/tex]

Since x = 9y:

   [tex]x=9y=9*\dfrac{1}{4} =\dfrac{9}{4}[/tex]

Let's find each side length:

  [tex]4x-4y=4(\dfrac{9}{4}) -4(\dfrac{1}{4} )=9-1=8\\\\2-2y=2-2(\dfrac{1}{4} )=2-\dfrac{1}{2} =\dfrac{3}{2} =1.5\\\\3x+5y=3(\dfrac{9}{4})+5(\dfrac{1}{4} )=\dfrac{27}{4} +\dfrac{5}{4} =\dfrac{32}{4}=8\\ \\x-3y=\dfrac{9}{4} -3(\dfrac{1}{4})=\dfrac{9}{4}-\dfrac{3}{4} =\dfrac{6}{4} =1.5[/tex]

Let's add up the side length to find the perimeter:

 [tex]\text{Perimeter}=8+1.5+8+1.5=9.5+9.5=19[/tex]

Answer: 19

The ratio of x to y is given by 3 : 1

The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.

The proportional equation is given as y ∝ x

And , y = kx where k is the proportionality constant

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Given data ,

Let the proportion be represented as A

Now , the value of A is

6 ( y + 3 ) = 2 ( x + 9 )

On simplifying , we get

6y + 18 = 2x + 18

Subtracting 18 on both sides , we get

6y = 2x

Divide by 2 on both sides , we get

x = 3y

Divide by 3 on both sides , we get

x/y = 3/1

Therefore , the proportion is x : y : : 3 : 1

Hence , the ratio is 3 : 1

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Step-by-step explanation:

(a)

If m is an odd integer, then m + 1 is an even integer.

m (odd integer) m + 1 (even integer)

1 2

3 4

5 6

... ...

Proof:

Suppose m is an odd integer. We can write m as 2n + 1 for some integer n. Then,

m + 1 = (2n + 1) + 1 = 2n + 2

Since 2n + 2 is clearly an even integer, it follows that m + 1 is an even integer if m is an odd integer.

(b)

If x is an even integer and y is an odd integer, then x + y is an odd integer

x (even integer) y (odd integer) x + y (odd integer)

0 1 1

2 3 5

4 5 9

... ... ...

Proof:

Suppose x is an even integer and y is an odd integer. We can write x as 2n and y as 2m + 1 for some integers n and m. Then,

x + y = 2n + (2m + 1) = 2(n + m) + 1

Since n + m is clearly an integer, it follows that x + y is an odd integer if x is an even integer and y is an odd integer.

(c)

If m is an even integer, then 3m^2 + 2m + 3 is an odd integer.

m (even integer) 3m^2 + 2m + 3 (odd integer)

0 3

2 27

4 99

... ...

Proof:

Suppose m is an even integer. We can write m as 2n for some integer n. Then,

3m^2 + 2m + 3 = 3(2n)^2 + 2(2n) + 3 = 12n^2 + 4n + 3

Since 12n^2 + 4n + 3 is clearly an odd integer, it follows that 3m^2 + 2m + 3 is an odd integer if m is an even integer.

Answer:

At point B rotate ABC clockwise 90 degrees.  Translate the figure two units to the left.  At point B dilate the figure with a scale factor of 2.

Step-by-step explanation:

P(A) = 1/2, P(B) = 3/4, and P(A and B) = 3/8, where A is the event that the first student is a boy and B is the event that the second student is a girl.

The total number of possible outcomes in this scenario is 4C2, which is equal to 6. These outcomes are AB, AC, AD, BC, BD, and CD, where A represents Franklin and B, C, and D represent Sandra, Marta, and Jane, respectively.

The probability that the first student is a boy is P(A) = 1/2, since there are two boys and four students total.

The probability that the second student is a girl is P(B) = 3/4, since there are three girls and four students total.

The probability that the first student is a boy and the second student is a girl is P(A and B) = 1/2 x 3/3 = 3/8, since the probability of the first student being a boy is 1/2 and the probability of the second student being a girl is 3/4 (after one girl has already been chosen as the first student).

Therefore, the probability that the first student is a boy is 1/2, the probability that the second student is a girl is 3/4, and the probability that the first student is a boy and the second student is a girl is 3/8.

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For square, Functions will be A=s², s=√A, s=√(d²/2), d=√2s², p=4s, s=p/4, A=(p/4)², d=√2A.

What exactly is a function?

A function is defined as a relationship between a group of inputs that each have one output. A function is a connection between inputs in which each input is associated to exactly one output. Every function has a domain and a co-domain, as well as a range. In general, a function is denoted as f(x), where x represents the input. A function's generic representation is y = f. (x).

In mathematics, there are several types of functions. Some examples include:

When there is a mapping for a range for each domain between two sets, this is referred to be an injective function or a one to one function.

Surjective functions, also known as Onto functions, are used when more than one element is transferred from domain to range.

Polynomial function: A function made up of polynomials.

Inverse Functions: A function that may be used to inverse another function.

Now,

As given square with side of length s, diagonal of length d, perimeter P, and area A.

and Area=side²

Perimeter=4*side

diameter²=side²+side²

then A=s² and s=√A,  s=√(d²/2) and d=√2s²,  p=4s and s=p/4, A=(p/4)², d=√2A.

Hence,

            For square, Functions will be A=s², s=√A, s=√(d²/2), d=√2s², p=4s, s=p/4, A=(p/4)², d=√2A.

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A node is a point of no displacement in a standing wave. Therefore, if point P does not move, it is most likely located at a node. the points along the wave that experience maximum displacement are called antinodes.

A node is a point of no displacement in a standing wave, meaning that if the stretched string of the apparatus represented is made to vibrate, point P will not move. Point P is most likely located at a node as it experiences no displacement. A standing wave is created when two waves combine and the resulting wave is stationary. The points along the wave that experience no displacement are called nodes, and the points along the wave that experience maximum displacement are called antinodes. Nodes can be found at points that are integral multiples of half the wavelength of the wave. Therefore, it can be concluded that point P is at a node since it does not move when the string is made to vibrate.

The complete question is :

When the stretched string of the apparatus represented below is made to vibrate, point p does not move. Point p is most probably at the location of _____.

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The value of cos (2x) is 7/25.

Trigonometric functions are defined as the real functions which are simply the functions of an angle of a triangle. They are basically the periodic functions which relate an angle in a right angled triangle to the ratios of the length of two sides.

Given that,

sin x = -3/5 and cos (x) < 0

We have a trigonometric formula,

cos (2x) = 1 - 2 sin²(x)

Substituting the values given,

cos (2x) = 1 - 2 × (-3/5)²

             = 1 - (2 × 9/25)

             = 1 - 18/25

             = 7/25

Hence the value of cos (2x) is 7/25.

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To find a common denominator for 7/8 and 13/16, we can find the least common multiple (LCM) of 8 and 16, which is 16.

LCM's meaning ?

lowest common factor

Describe LCM. Least Common Multiple is a mathematical term. The smallest number that is a multiple of both of two numbers is called the least common multiple.

7/8 can be written as 7 * (2/2) / 8 = 7 * 2 / 16 = 7/16

13/16 is already in the form of a fraction with a denominator of 16, so no further modification is needed.

So, 7/8 can be written as 7/16 and 13/16 can be written as 13/16.

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The probabilities of the random sample of 223 students are solved

P ( male ) = 0.475

P ( female ) = 0.525

P ( male | pet ) = 0.538

P ( female | no pet ) = 0.547

The probability that an event will occur is measured by the ratio of favorable examples to the total number of situations possible

Probability = number of desirable outcomes / total number of possible outcomes

The value of probability lies between 0 and 1

Given data ,

Let the total number of students be = 223 students

Let the total number of male students be = 49 + 57 = 106 students

Let the total number of female students be = 64 + 53 = 117 students

Now , the equation will be

Let the number of male students who own a pet = 57 students

Let the number of male students who does not own a pet = 49 students

And ,

Let the number of female students who own a pet = 53 students

Let the number of female students who does not own a pet = 64 students

The probability of choosing a male student P ( male ) = number of male students / total number of students

The probability of choosing a male student P ( male ) = 106 / 223

The probability of choosing a male student P ( male ) = 0.475

And ,

The probability of choosing a female student P ( female ) = number of male students / total number of students

The probability of choosing a female student P ( female ) = 117 / 223

The probability of choosing a female student P ( female ) = 0.525

And ,

Probability of choosing a male student who owns a pet P ( male | pet ) = number of male students who own a pet / number of male students

P ( male | pet )  = 57 / 106

P ( male | pet ) = 0.538

The probability of choosing a female student who does not own a pet is P ( female | no pet ) = number of female students who does not own a pet / number of female students

P ( female | no pet ) = 64 / 117

P ( female | no pet ) = 0.547

Hence , the probabilities are solved

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Answer:[tex]\frac{32\pi }{3} cm^{3}[/tex]

Step-by-step explanation:

Since the diameter is 4 cm, we know the radius is 2 cm since diameter = 2 x radius. The formula for a sphere's volume is [tex]\frac{4\pi }{3}[/tex]×[tex]r^{3}[/tex], so by plugging in 2 we get  [tex]\frac{4\pi }{3}[/tex]×[tex]2^{3}[/tex] = [tex]\frac{4\pi }{3}[/tex] x 8 = [tex]\frac{32}{3}[/tex] [tex]\pi[/tex] [tex]cm^{3}[/tex]

The volume of the spherical boulder is 7234.56 cubic feet.

What is the diameter?

A line connecting the center and the circumference at its opposite ends is called the diameter. Its length is double that of the circle's radius.

The formula for the volume of a sphere is [tex]V=\frac{4}{3}\pi r^{3}[/tex].

Given the diameter of the sphere is 24 feet.

therefore radius is equal to 12 feet.

The volume of the sphere is equal to

[tex]V=\frac{4}{3}\pi (12)^{3} \\V=\frac{4}{3} *3.14*1728\\V=7234.56[/tex]

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The equation of the line secant to the cubic equation y = x³ - x + 4 is equal to y = 3 · x + 4.

In this problem we find the case of a cubic equation that is intersected twice by a line, that is, a secant line. According to analytical geometry, lines are described by equations of the form:

y = m · x + b

Where:

Where the slope of the line is determined by secant line formula:

m = Δy / Δx

First, determine the slope of the secant line:

x = - 2

y = (- 2)³ - (- 2) + 4

y = - 2

x = 2

y = 2³ - 2 + 4

y = 10

m = [10 - (- 2)] / [2 - (- 2)]

m = 3

Second, calculate the intercept of the linear function:

b = y - m · x

b = 10 - 3 · 2

b = 10 - 6

b = 4

Third, write the equation of the secant line:

y = 3 · x + 4

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√(a² + b²)²  - 4a²b²cos²θ is the product of the lengths of the two diagonals is given by the expression.

What is a mathematical expression?

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. This mathematical operation may be addition, subtraction, multiplication, or division.

                                An expression's structure is as follows: Number/variable, Math Operator, Number/Variable is an expression.

we have AB as a, AD as b and the angle between them is theta.

So using the cosine rule, we have

  BD = √a² + b² - 2abcosθ

So now consider the triangle ABC

Here AB is a, BC is b and the angle is 180-theta

So using cosine rule, we get AC as

   AC = √a² + b² - 2abcosθ( 180 - θ )

    AC = √a² + b² - 2ab(-cosθ )

    AC = √a² + b² - 2abcosθ

Now we have the two diagonals AC and BD. So multiplying, we get

 AC × BD = √a² + b² + 2abcosθ × √a² + b² - 2abcosθ

Simplifying, we get

AC × BD = √(a² + b² + 2abcosθ) × (√a² + b² - 2abcosθ)

AC × BD = √(a² + b²)² - (2abcosθ)²

AC × BD = √(a² + b²)²  - 4a²b²cos²θ

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The benefits of using quadratic models in real-world applications include: -They are versatile and can be used for a variety of problems.

Researchers may find the quadratic model to be a useful data analytic approach for helping them identify the combined impacts of achievement goals on academic accomplishment.

We anticipate that future studies on the impact of academic achievement goals will frequently use the quadratic model to analyze their data.

Situations that can be approximated by quadratic functions include tossing a ball, firing a cannon, jumping off a platform, and hitting a golf ball.

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Answer:

Greatest common factor for 14/16 is 2

Greatest common factor for 3/12 is 3

Greatest common factor for 16/28 is 4

So to simplify 14/16 it is 7/8

To simplify 3/12 it is 1/4

To simplify 16/28 it is 4/7

Step-by-step explanation:

[tex]y=\frac{2x\pm\sqrt{5x^2-(4x^2-16x+4\bar{c})} }{2(1)}[/tex]Solve the given DE, [tex](y-x)\frac{dy}{dx} =y-x+2[/tex].

Rewriting,

=> [tex](y-x)\frac{dy}{dx} =y-x+2[/tex]

=> [tex](y-x)dy =(y-x+2)dx[/tex]

=> [tex]-(y-x+2)dx+(y-x)dy =0[/tex]

=> [tex](-y+x-2)dx+(y-x)dy =0[/tex]

Check to see if this is an exact DE by taking the partial derivative of M with respect to y and N with respect to x.

[tex]M=(-y+x-2)dx[/tex]

=> [tex]M_{y} =-1[/tex]

[tex]N=(y-x)dy[/tex]

=> [tex]N_{x}=-1[/tex]

[tex]M_{y} =N_{x}[/tex], so this is an exact DE. Now integrate M with respect to x and N with respect to y.

[tex]\int\ ({-y+x-2)} \, dx[/tex]

=>[tex]-xy+\frac{x^2}{2}-2x[/tex]

[tex]\int\ ({y-x)} \, dy[/tex]

=> [tex]=\frac{y^2}{2} -xy[/tex]

So we can say the solution to the given DE is, [tex]\frac{x^2}{2}+\frac{y^2}{2}-xy-2x=c[/tex].

Anna must sell at least approximately 1.845 books to make a profit.

What is the linear equation?

A linear equation is an algebraic equation of the form y=mx+b. where m is the slope and b is the y-intercept.

We can find the minimum number of books Anna must sell to make a profit by setting the revenue equal to the cost and solving for x.

That is, we want to find the value of x where R = C:

6.56x = 10.063 + 1.09x

5.47x = 10.063

x = 10.063 / 5.47

x = approximately 1.845 books

Hence, Anna must sell at least approximately 1.845 books to make a profit.

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4966.5

Step-by-step explanation:

Starting at finding out how the population will increase in 3 years we take 3 and divide it by 4. This produces an increase of 75% every 3 years. If we multiply 2838 by 75% we get 2128.5. If we add it back to 2838, we get 4966.5

Step-by-step explanation:

The equation 3x + y = 0 can be rearranged to y = -3x. This means that the graph of the equation is a straight line with slope -3 and y-intercept of 0.

The three points that are in the solution set are (-2,6), (-5,5), and (3,-9). Plotting these points on a coordinate plane and connecting them with a straight line, we can see that they all lie on the same line with slope -3:

[Graph not shown here, but a straight line with three points (-2, 6), (-5, 5), (3, -9) can be plotted with a slope of -3 and y-intercept of 0.]

From the graph, we can see that the line passes through the origin (0,0), which means that the y-intercept is 0. The slope of the line is -3, which means that the line has a negative slope and is downward sloping. The line separates the plane into two half-planes, one above the line and one below the line. The points above the line have y-coordinates greater than 0 and the points below the line have y-coordinates less than 0.

A quadratic function with x-intercepts as (-2,0) and (1,0) and y-intercept as (0,-2) is y = x² + x - 2.

What is a quadratic function?

A polynomial function with one or more variables, where the largest exponent of the variable is two, is referred to as a quadratic function. It is also known as the polynomial of degree 2 since the greatest degree term in a quadratic function is of second degree.

It is given that the x intercepts of the quadratic function are (-2,0) and (1,0).

The y intercept of the quadratic function is (0,-2).

Let the equation of the given quadratic be y = ax² + bx + c.

As the given quadratic has x intercepts as (-2,0) and (1,0) and y intercept as (0,-2).

This implies that the quadratic function is passing through the points (-2,0), (1,0) and (0,-2).

So, the points (-2,0), (1,0) and (0,-2) must satisfy the equation of the quadratic function.

As the point (-2,0) satisfy the equation of the quadratic function -

y = ax² + bx + c

0 = a(-2)² + b(-2) + c

0 = 4a - 2b + c ..... (1)

As the point (1,0) satisfy the equation of the quadratic function -

y = ax² + bx + c

0 = a(1)² + b(1) + c

0 = a + b + c ..... (2)

As the point (0,-2) satisfy the equation of the quadratic function -

y = ax² + bx + c

-2 = a(0)² + b(0) + c

-2 = c ..... (3)

Substitute the value of c in equation (1) -

0 = 4a - 2b - 2

2 = 4a - 2b ...... (4)

Substitute the value of c in equation (2) -

0 = a + b - 2

2 = a + b ...... (5)

Multiply equation (5) by 2 -

4 = 2a + 2b ...... (6)

Add equation (4) and (6) -

2 + 4 = 4a - 2b + 2a + 2b

6 = 6a

a = 1

Substitute the value of a in equation (5) -

2 = 1 + b

b = 2 - 1

b = 1

The values are a = 1, b = 1 and c = -2.

Now substitute the value of a, b and c in the quadratic function.

y = ax² + bx + c

y = (1)x² + (1)x + (-2)

y = x² + x - 2

Therefore, the quadratic function is y = x² + x - 2.

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The annual deposits that needs to be made is for amount $2352.94.

What is annuity?

Annuity refers to an equal series of future cash flows which are received or paid periodically. Future value of annuity is the value of the annuity at the end of the series whereas present value is the value of the annuity at the beginning of the series.

The annuity value can be calculated as -

Future value of annuity = Annuity x (1 - (1 + Rate)^-Number of years) / Rate

The values are given as -

Future value of annuity = $17500.00

Annuity = Identical annual deposits

Rate = 4% = 0.04

Number of years = 9

Substitute the values into the equation -

17500 = Annuity x (1 - (1 + 0.04)^-9 ) / 0.04

Annuity identical deposits = 17500 x 0.04 / ((1.04)^-9)

Annuity identical deposits = 700 / (1 - 0.7025)

Annuity identical deposits = 700 / 0.2975

Annuity identical deposits = 2352.94

Therefore, the value is obtained as $2352.94.

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The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

R1 = 4, R2 = 5, R3 = 6

The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

The integral is given by:

∫ (R1 + R2 + R3) dA

where R1, R2, and R3 are the areas of the labeled regions in the figure.

By interpreting the integral in terms of areas, we can calculate the value of the integral. The integral evaluates to 11, which is the sum of the areas of the three regions (R1 + R2 + R3 = 4 + 5 + 6 = 11).

The complete question is :

Evaluate the integral below by interpreting it in terms of areas in the figure. The areas of the labeled regions are A = 3, B = 4, C = 5, and D = 6.

∫DBCA x dA

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