Show all steps to verify the trigonometric identity:
1 - 2cos²x = 2sin²x - 1

Please include all steps!!!​

2/5th of a number is 40% of that number.

In order to calculate the percentage of a given number that is equal to 2/5 of that number, we must first convert the fraction to percentage form, which is as follows:

= 2/3 * 100

= 0.4 * 100

= 40%

We will use the following example to validate the aforementioned process:

Assume that it is 50.

First, 2/5 of 50 equals 2/5 * 50, or 20.

Second, 40% of 50 equals 40/100 * 50 = 0.4 * 50 = 20.

The equality of the two results is evident.

2/5 of a number is therefore 40% of that number.

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The question is incomplete;

\dfrac25 start fraction, 2, divided by, 5, end fraction of a number is what percentage of that number?

Answer:

1) 117

2) 24

Step-by-step explanation:

Corresponding sides are in the same ratio for similar triangles

1)

For triangles MUL and WUV:

UW/UM = UV/UL

?/45 = 104/40

? = 45 x 104/40 = 117

2)
For triangles RST and RKH
RS/RK = RT/RJ

?/12 = 20/10 = 2

? = 2 x 12 = 24

The value of your investment at maturity at a 10% rate of return will be,

$8470.

Borrowers are required to pay interest on interest in addition to principal since compound interest accrues and is added to the accrued interest from prior periods.

We know the formula for compound interest is,

A = P(1 + r/100)ⁿ.

Where, A = amount, P = principle, r = rate, and n = time in years.

Given, An initial investment of $7,000 and when you choose to invest for 14 years you get a 10% rate of return.

We know, 10% = 10/100 = 0.1.

Therefore, The value of your investment at maturity will be,

y = 7000(1 + 0.1)².

y = 7000(1.1)².

y = 7000×1.21.

y = 8470.

So, The value of the maturity will be $8470.

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(a) Circumference of Earth in kilometers: 40,075.017 km

(b) Surface area of Earth in square kilometers: 510.1 million km^2

(c) Volume of Earth in cubic kilometers: 1.08 trillion km^3

To calculate these values, we use mathematical formulas that relate to the sphere shape of the Earth.

For (a), we use the formula for the circumference of a circle, C = 2 * pi * r, where r is the radius of the Earth in meters (6.37 x 10^6 m). We then convert this result to kilometers by dividing by 1000.

For (b), we use the formula for the surface area of a sphere, A = 4 * pi * r^2, where r is the radius of the Earth. Again, we convert this result to square kilometers.

For (c), we use the formula for the volume of a sphere, V = (4/3) * pi * r^3, where r is the radius of the Earth. We then convert this result to cubic kilometers by dividing by 10^9.

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The third column of the matrix is:

[tex]\begin{bmatrix}1 \\5 \\7\end{bmatrix}[/tex]

The third column of a matrix can be calculated without computing the other two columns by using the formula:

A_{ij} = (A_{i1} - A_{i2}) \times \frac{j - 1}{2} + A_{i2}

Where A_{ij} is the element in the ith row and jth column of the matrix, A_{i1} is the element in the ith row and 1st column of the matrix and A_{i2} is the element in the ith row and 2nd column of the matrix.

To calculate the third column of a matrix, we will first calculate the difference between the first two columns. For example, if we have the following matrix:

[tex]\begin{bmatrix}1 & 4 & ? \\7 & 2 & ? \\3 & 8 & ?\end{bmatrix}[/tex]

The difference between the first two columns is 3 (4 - 1 = 3). Now we can use the formula to calculate the third column. For the first row, we have A_{11} = 1, A_{12} = 4, and j = 3 (the third column). Plugging these values into the formula, we get:

A_{13} = (1 - 4) \times \frac{3 - 1}{2} + 4 = 1

For the second row, we have A_{21} = 7, A_{22} = 2, and j = 3 (the third column). Plugging these values into the formula, we get:

A_{23} = (7 - 2) \times \frac{3 - 1}{2} + 2 = 5

For the third row, we have A_{31} = 3, A_{32} = 8, and j = 3 (the third column). Plugging these values into the formula, we get:

A_{33} = (3 - 8) \times \frac{3 - 1}{2} + 8 = 7

Therefore, the third column of the matrix is:

[tex]\begin{bmatrix}1 \\5 \\7\end{bmatrix}[/tex]

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The equation we will use is Equation 1: 2x − 3y = 11. The values of x and y are 1 and -3.

A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.

Given;

Equation 1: 2x − 3y = 11

Equation 2: x + 2y = −5

Now,

Equation 2: x + 2y = −5

x=-5-2y

Substituting the value of x in equation 1

2(-5-2y)-3y=11

-10-4y-3y=11

-10-7y=11

-7y=11+10

-7y=21

y=-3

Substituting the value of y in equation 2

x+2*-3=-5

x=-5+6

x=1

Therefore, the solution of the system of equations will be x=1,y=-3

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The expression is factorized to give x(x+ 12) + 8( y + 12)

From the information given, we have that the expression as;

(x+y)²+8(x+y)+12

Now, expand the bracket, we get;

(x + 2) (x+2) + 8x + 8y + 12

expand the squared bracket, we get;

x² + 2x + 2x + 4 + 8x + 8y + 12

Now, collect like terms

x² 2x + 2x + 8x + 8y + 4 + 12

Add the collected like terms, we get;

x² + 12x + 8y + 16

Now, let's factorize in airs by grouping, we get;

(x² + 12x) + (8y + 16)

Factor the common terms

x(x+ 12) + 8( y + 12)

Hence, the expression is x(x+ 12) + 8( y + 12)

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The complete coordinates are (8, 0), (0, -8), (-8, 0) and (0, 8)

From the question, we have the following parameters that can be used in our computation:

x^2 + y^2 = 64

The above is the equation of a circle

Express 64 as 8^2

So, we have

x^2 + y^2 = 8^2

When y = 0, the possible values of x are

x = -8 and x = 8

When x = 0, the possible values of y are

y = -8 and y = 8

This means that the complete coordinates are (8, 0), (0, -8), (-8, 0) and (0, 8)

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If line m and n are parallel, the value of x is determined as 10.

The value of x is determined by applying the following rules of parallel lines as shown below.

The corresponding angles of the parallel lines are equal,

the corresponding angles are given as;

8x + 50 = 130

now solve for x, by making x the subject of the formula.

8x = 130 - 50

8x = 80

divide through by 8

8x / 8 = 80 / 8

x = 10

Thus, the value of x is function of the corresponding angles.

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If statement p is true but statement q is false, then the following compound statements are true -

Option B: p ∨ q

Option D: q → p

What is compound statement?

A compound statement is a collection of two or more statements joined together using terms like "or," "and," "if then," and "only if." Each component statement of a compound statement can be determined to be true or untrue with clarity.

If the truth value of the statement p is 'True' and truth value of statement q is 'False',

p ∧ q → False [Conjunction of both the statements will be true only when both the statements are True]

p ∨ q → True [Disjunction of two statements will be false only when both the statements are false]

(p → q) → False [p is same as q]

(q → p) → True [Implication of both the statements is false only when p is true and q is false otherwise true]

(p ↔ q) → False [Bi-conditional statements are true if and only when both the statements are true]

(q ↔ p) → False [Bi-conditional statements are true if and only when both the statements are true]

Therefore, the compound statements p ∨ q and (q → p) is true.

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The study of mathematical logic help you in your everyday life in the following way.

Mathematical logic is a branch of mathematics that deals with the principles of reasoning and makes statements based on deductive reasoning.

It is not just a subject that helps you ace your math exams but also a useful tool in your everyday life.

Understanding the basic principles of logic can help you make better decisions, solve problems more effectively, and communicate your ideas more clearly.

Logic helps you analyze information, evaluate arguments and make informed decisions.

In everyday life, you often encounter situations where you have to make choices based on information that is incomplete or ambiguous.

Furthermore, logic can also help you improve your critical thinking skills. By understanding how to reason and argue logically, you can more easily identify and evaluate flawed arguments, understand the relationships between different ideas and reach more accurate conclusions.

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The distributive property to factor out the greatest factor 24+18t is 6(4+3t).

The distributive property to factor out the greatest common factor 24+18t.

The Distributive Property is an algebraic property that is used to multiply a single value and two or more values within a set of parenthesis. The distributive Property States that when a factor is multiplied by the sum/addition of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition operation. This property can be stated symbolically as:

Well, some of the common factors that 24 and 18 shares are 2, 3, and 6.

The greatest factor is 6.

⇒ [tex]\frac{24}{6}[/tex]=4

⇒ [tex]\frac{18t}{6}[/tex]=3

Because 6x4=24 and 6 x 3t= 18t

Therefore, the distributive property to factor out the greatest factor 24+18t is 6(4+3t).

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The solution to the system of linear equations is: x = 5, y = (6x + 7 - 10/4)/(-5) = -4/5, z = -5/4.

To solve the system of linear equations using elimination, we can manipulate the coefficients of the variables to make one of the variables zero in one of the equations. Then, we can use this equation to eliminate that variable from the other equations.

Starting with the first two equations:

6x - 5y - 4z = -7

5x - 4y - 4z = -5

We can multiply the first equation by 5 and the second equation by 6 to get rid of the coefficients of x in both equations:

30x - 25y - 20z = -35

30x - 24y - 24z = -30

Now, we can subtract the first equation from the second equation to eliminate x:

0y - 4z = 5

So, z = -5/4

Next, we can use this value of z to solve for y in either of the first two equations:

6x - 5y - 4(-5/4) = -7

Expanding the right side:

6x - 5y + 10/4 = -7

And solving for y:

y = (6x + 7 - 10/4)/(-5)

Finally, we can use the value of y and z to solve for x in any of the first two equations:

6x - 5((6x + 7 - 10/4)/(-5)) - 4(-5/4) = -7

Expanding and solving for x:

6x - (6x + 7 - 10/4)/5 - 10/4 = -7

Multiplying both sides by 5:

30x - 6x - 7 + 2 = -35

And solving for x:

x = 5

Finally, we can use the values of x, y, and z to check the third equation:

3y + 6z = 6

Substituting the values we found:

3((6x + 7 - 10/4)/(-5)) + 6(-5/4) = 6

Expanding and solving:

The left side is equal to 6 and the right side is equal to 6, so the solution is consistent.

Therefore, the solution to the system of linear equations is:

x = 5, y = (6x + 7 - 10/4)/(-5) = -4/5, z = -5/4

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The sample space for choosing one card is, S={I, L, L, M, P, T, U, Y}

A sample space is a set of potential results from a random experiment. The letter "S" is used to denote the sample space. Events are the subset of possible experiment results. Depending on the experiment, a sample area could contain a variety of results.

Given that,

A set of eight cards were labeled with M, U, L, T, I, P, L, Y.

Here, the sample space is;

{ S, U, B, T, R, A, C, T}

Now, Elements in order is,

⇒ S = {I, L, L, M, P, T, U, Y}

Therefore, the sample space for the given cards is,

S = {I, L, L, M, P, T, U, Y}

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Answer: I think the answer would be C

Answer:

X=5 Cars

Y=6 Vans

Step-by-step explanation:

Let's assume the number of cars used is x and the number of vans used is y.

We can set up two equations based on the information given:

x + y = 11 (the total number of vehicles used)

5x + 8y = 73 (the total number of students that went on the trip)

We can use substitution to solve for one of the variables in terms of the other:

y = 11 - x

5x + 8(11 - x) = 73

5x + 88 - 8x = 73

-3x = -15

x = 5

So the class used 5 cars and 11 - 5 = 6 vans.

The number of bricks needed to border the circular pool is 6.28r.

The circumference of a circle is the distance around the circle which is 2πr.

The diameter of a circle is the largest chord that passes through the center of a circle it is 2r.

Given, Mr. Cortez is putting a brick border around his circular pool.

So, He is boarding the circumference of the circular pool which has a distance of 2πr.

Now, The bricks are 1 foot in length.

Therefore, The number of bricks needed is,

= 2πr/1.

= 2πr.

= 6.28r.

According to the radius number of bricks can be determined.

Q. Mr. Cortez is putting a brick border around his circular pool. The bricks are 1 foot in length. How many bricks does Mr. Cortez need to buy to build the border? Use 3.14 for π.

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use the definition of big o notation to find the constants c, no which show that t(n) is o(f(n)) then  5^n+25 is not = 5^n

The main purpose of the computational complexity is to categorize the algorithms about their show. And we can represent the T(n), which is the time function with the help of the big O notation, for showcasing an algorithm runtime complexity. As an example,

T(n) = O(n3)

proves that the algorithm has the cubic time complexity.

Now let's define Bio O, and from here we will get our answer. For any function which is monotonic in nature, f(n) and g(n) from the +ve integers to the +ve integers, we assume that f(n)= O(g(n)) when exist a c>0 which is the constant, and n0 greater than 0, such that

f(n) < c * g(n), for each n>=n0

i) if O (5^n)= 4^n + 25

O (f) should be = (20^n)/f + 25

Then constant

O(5^n)= (20^n)/(5^n) + 25 = (20/5)^n +25=4^n+25

ii) O (4^n)=(20^n)/(4^n) + 25 = (20/4)^n +25=5^n+25

And 5^n+25 is not = 5^n

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complete question is

Use the definition of O (f) to show that 4^n + 25 is O (5^n) (ii) show that 5^n is not O (4^n)

The slope of the line perpendicular to y = x is -1. The equation of a line in slope-intercept form is given by y = mx + b, where m is the slope and b is the y-intercept. Substituting m = -1 and using the point (2, 2), we have:

y = -x + b

2 = -2 + b

b = 4

So the equation of the line passing through (2, 2) and perpendicular to y = x is:

y = -x + 4

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The value of the integral expression is -1/2ln(9)

From the question, we have the following parameters that can be used in our computation:

Integral of 4/(9 - 8x) from 2 to 9

Using a graphing calculator as a guide, we have the following:

Integrand = -ln(|8x - 9|)/2 from 2 to 9

Substitute the known values in the above equation, so, we have the following representation

Integrand = -ln(|8 * 9 - 9|)/2 + ln(|8 * 2 - 9|)/2

Evaluate

Integrand = -1/2(ln(63/7)

Evaluate the quotient

Integrand = -1/2ln(9)

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The Pythagorean Theorem can be used to solve for the missing sides of a triangle when given two other sides

a. The missing components that can be solved using the given values are the missing sides of the triangle (side b and side c).

b. To find the missing side, we can use the Pythagorean Theorem, which states that a^2 + b^2 = c^2, where a and b are the two sides of the triangle, and c is the hypotenuse. Therefore, we can solve for c by taking the square root of (a^2 + b^2).

The Pythagorean Theorem can be used to solve for the missing sides of a triangle when given two other sides. To do this, we take the square root of the sum of the squares of the two given sides, which gives us the length of the hypotenuse.

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The original sum of money borrowed would be $ 2246.51.

A financial concept called future value (FV) determines the worth of an asset by using projected factors like future interest rates or cashflows. An investor could find it interesting to know how much, given a projected rate of return, their investment might be in five years. Future value is the idea that an investment will be worth more in the future if its current value is multiplied by anticipated growth.

Interest = $ 96.60

Interest is compounded 6 times in a year; n = 6

time = 1 year ; Rate of interest (r) = 4.2 %

Interest = Future Value - Principal Value

Future Value = Principal [tex]$\cdot\left(1+\frac{r}{100 \times n}\right)^{n \cdot t}$[/tex]

Substituting this value in equation (1), We get

Interest = Principal [tex]$\cdot\left(1+\frac{r}{100 \times n}\right)^{n \cdot t}-$[/tex]Principal

96.60 = Principal [tex]$\left[\cdot\left(1+\frac{4.2}{100 \times 6}\right)^{6 \cdot 1}-1\right]$[/tex]

Principal = $ 2246.51

Therefore, the original sum of money borrowed would be $ 2246.51.

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Therefore, in order for Joey to earn the same hourly wage as Blake, she will have to labor for 8 six-month periods or 4 years.

We'll refer to the quantity of six-month intervals as x. Blake's hourly wage can be represented as 16 + 0.50x, and Joey's hourly wage may be represented as 12 + 1.50x after x intervals. We may set the two expressions equal to one another and solve for x to determine when Joey's salary will match Blake's salary:

12 + 1.50x = 16 + 0.50x \s0.50x = 4 \sx = 8

Joey will therefore need to work for 8 six-month periods, or 4 years, in order to make the same hourly rate as Blake.

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Solving the provided question, we can say that the given exponent is

[tex]32 = 2^{ 3x - 1 }[/tex] => 64 = [tex]2^{3x}[/tex] => [tex]2^{3(2)} = 2^{3x}[/tex] on comparing both side, we have x = 2

Exponentiation, often known as "b raised to the nth power," is a mathematical operation denoted by the symbol bn that involves two numbers: a base number, b, and an exponent, or power, n. Exponents show how many times a number has been multiplied by itself. As an illustration, 2-3 (written as 23) denotes: 2 x 2 x 2 = 8. 23 is not equivalent to 2 + 3 = 6. Recall that an integer raised to the power of one equals itself. A approach to express huge numbers as powers is through the use of exponents. Exponent, then, is the quantity that indicates how many times a number has been multiplied by itself. As an illustration, 6 is multiplied by 4 to provide 6 x 6 x 6 x 6. It may be expressed as 64. 4 is where

the given exponent is

[tex]32 = 2^{ 3x - 1 }[/tex]

64 = [tex]2^{3x}[/tex]

[tex]2^{3(2)} = 2^{3x}[/tex]

on comparing both side, we have

x = 2

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Answer:

(x - 7)(x + 8)

Step-by-step explanation:

x² + x - 56
Multiples of 56 whose difference is 1: 7, 8
So, splitting the equation:
x² + 8x - 7x - 56
(x - 7)(x + 8)

The factored form of a quadratic expression is a way to rewrite the expression which shows the sums of products.

general form: [tex]y= a(x-r)(x-s)[/tex]

when the a value is equal to zero:

[tex]y=x^2+x-56[/tex]

[tex]y=(x+8)(x-7)[/tex]

The volume of the solid is 0 cubic units.

Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.

There are two main types of integration: definite and indefinite integration. Definite integration involves finding the exact area between a curve and the x-axis over a specified interval. Indefinite integration involves finding the general antiderivative of a function, which is a function that, when differentiated, will yield the original function.

The process of integration can be represented symbolically as ∫ (the integral symbol) and the result of an integration is typically represented as an antiderivative. The fundamental theorem of calculus states that differentiation and integration are inverse operations, so the derivative of an antiderivative is the original function.

The volume of the solid is given by the definite integral:

V = ∫_[tex]{y=2}^{1} A(y) dy[/tex]

Where A(y) is the area of the cross-section at y. Since the cross-sections are squares, we know that A(y) = [tex]y^{2}[/tex].

So, the volume is given by:

V = ∫_[tex]{y=2}^{1} y^{2} dy = [\frac{y^{3} }{3} ]_{y=2}^{y=1} = (\frac{8}{3} - (\frac{2^{3} }{3}= (\frac{8}{3}-\frac{8}{3} ) =0[/tex]

The volume of the solid is 0 cubic units.

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The measure of each interior angle of the regular polygon is equal to 150°.

In Euclidean geometry, a regular polygon simply refers to a form of polygon which has all of its sides and interior angles being equal.

In Geometry, the sum of the interior angles of both a regular and irregular polygon is given by this formula:

Sum of interior angles = 180 × (n - 2)

Where:

n represents the number of sides of a regular polygon.

Similarly, the measure of each interior angle of a regular polygon can be calculated by using this mathematical expression:

Interior angles = [180 × (n - 2)]/n

Interior angles = [180 × (12 - 2)]/12

Interior angles = 180 × (10)/12

Interior angles = 1800/12

Interior angles = 150°

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Answer: Acute Triangles

The population of the world after 1 year will be 6,974,400,000 + 1.2% of 6,974,400,000 = 7,058,092,800

A figure or ratio stated as a fraction of 100 is called a percentage. Although the abbreviations "pct.", "pct.", and occasionally "pc" are also used, the percent symbol, "%," is frequently used to indicate it. A % is a number without dimensions and without a standard measurement.

By dividing the value by the entire value and multiplying the result by 100, one may get the percentage. The percentage calculation formula is (value/total value)100%.

The word "percentage" means "out of 100." In mathematics, percentages are used, together with fractions and decimals, to denote parts of a whole. The total is split into 100 equal pieces to calculate percentages.

Total population of the world is 6,974,400,000

The population is incresing at a 1.2% rate annually

The population of the world after 1 year will be 6,974,400,000 + 1.2% of 6,974,400,000 = 6,974,400,000 + 836928000 = 7,058,092,800

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The area of the shaded portion is area of the outer circle - area of the inner circle = 1017.36 - 379.94 = 637.42 yards²

All points on a plane that are at a specific distance from a specific point, the center, form a circle. In other words, it is the curve that a moving point in a plane draws to keep its distance from a specific point constant.

The following are some of the circle's crucial characteristics:

The radius of the inner circle = 22 /2 = 11 yard = r1

The distance between the inner cirle and outer circle = 7 yards

The radius of the outer circle = 18 yards = r2

The Area of the outer circle = π*r2² = 3.14 * 18² = 1017.36 yards²

The Area of the inner circle = π*r1² = 3.14 * 11² = 379.94 yards²

The area of the shaded portion is area of the outer circle - area of the inner circle = 1017.36 - 379.94 = 637.42 yards²

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The coefficient of determination (r-squared) provides an indication of how well the regression model fits the data and the value of r is  0.3574

In this example, the r-squared value is 12.78%, which means that 12.78% of the variation in verbal SAT scores can be explained by the variation in math SAT scores. The least squares regression equation is yhat = 383.3 + 0.3489x, where yhat is the predicted verbal score and x is the math score.

The coefficient of correlation (r) provides a measure of the strength of the linear relationship between two variables. In this example, the coefficient of correlation (r) can be calculated from the coefficient of determination (r-squared) as follows: r = √r-squared = √12.78% = 0.3574. This indicates that there is a weak linear relationship between verbal and math SAT scores.

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