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The solutions to the original equation are x = 2nπ, where n is an integer.

A trigonometric identity is an equation that is true for all values of the variables within the domain of the functions involved.

We can use the trigonometric identity cos(2x) = cos²(x) - sin²(x) to simplify the left-hand side of the equation:

cos(2x)cos(x) - sin(2x)sin(x) = (cos²(x) - sin²(x))cos(x) - 2sin(x)cos(x)sin(x)

Simplifying further using the identity sin(2x) = 2sin(x)cos(x), we get:

(cos²(x) - 3sin²(x)cos(x)) + sin(2x) = 1

Substituting sin(2x) = 2sin(x)cos(x), we get:

cos³(x) - 3sin²(x)cos(x) + 2sin(x)cos(x) - 1 = 0

We can factor this expression as follows:

(cos²(x) - 1)(cos(x) - 2sin(x) + 1) = 0

The first factor has solutions cos(x) = ±1, which means x is an integer multiple of π/2. However, these solutions do not satisfy the original equation, since the left-hand side is always equal to zero.

The second factor gives us the equation cos(x) - 2sin(x) + 1 = 0. We can rewrite this as:

cos(x) + 1 = 2sin(x)

Using the identity sin²(x) + cos²(x) = 1, we can square both sides of this equation to get:

1 + 2cos(x) + cos²(x) = 4sin²(x)

Substituting sin²(x) = 1 - cos²(x), we get:

5cos²(x) - 4cos(x) - 3 = 0

This quadratic equation can be factored as:

(5cos(x) + 3)(cos(x) - 1) = 0

The first factor gives us the solution cos(x) = -3/5, but this is not a valid solution since the cosine function only takes values between -1 and 1.

The second factor gives us the solution cos(x) = 1, which means x is an integer multiple of 2π.

Therefore, the solutions to the original equation are:

x = 2nπ, where n is an integer.

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Answer: If the height of seven and three fourths yards and a base of 20 yards, then the area of the right triangle is 77.5 square yards.

To calculate the area of a right triangle, we can use the formula:

Area = (base * height) / 2

where the base is one of the sides of the triangle, and the height is the perpendicular distance from that side to the opposite vertex.

In this case, the height of the right triangle is 7 and 3/4 yards, and the base is 20 yards. Plugging these values into the formula, we get:

Area = (20 * 7.75) / 2 = 155 / 2 = 77.5 square yards

In summary, to calculate the area of a right triangle, we can use the formula that relates the base and height to the area. In this case, we needed to use the given values of the height and base of the right triangle to compute the area in square yards.

The resulting area represents the amount of surface inside the triangle and is useful in many applications, such as geometry, physics, and engineering.

Step-by-step explanation:

Hope this helps! =D

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There are a total of 5 + 4 + 2 + 2 + 1 = 14 possible combinations for Jacob's lock.

The first digit of Jacob's lock combination can be any of the factors of 84: 1, 2, 3, 4, 6, 7, or 12.

If the first digit is 1, then the second digit can be any of the factors of 84/1 that are greater than 1 and less than 10 (to avoid repeating digits): 2, 3, 4, 6, or 7. There are 5 choices for the second digit.

If the first digit is 2, then the second digit can be any of the factors of 84/2 that are greater than 2 and less than 10: 3, 4, 6, or 7. There are 4 choices for the second digit.

If the first digit is 3, then the second digit can be any of the factors of 84/3 that are greater than 3 and less than 10: 4 or 6. There are 2 choices for the second digit.

If the first digit is 4, then the second digit can be any of the factors of 84/4 that are greater than 4 and less than 10: 6 or 7. There are 2 choices for the second digit.

If the first digit is 6, then the second digit can only be 7. There is 1 choice for the second digit.

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Answer:

Step-by-step explanation:

Using intersecting secant theorem, the value of x is approximately 6

The value of the variable is solved using intersection secant theorem

The theorem gives the formula applied as follows

(6 + 6) * 6 = (5 + x) * x

Starting with the left side of the equation:

(6 + 6) * 6 = 12 * 6 = 72

Now, let's simplify the right side of the equation:

(5 + x) * x = 5x + x^2

Substituting these simplified expressions back into the original equation, we get:

72 = 5x + x^2

This is now a quadratic equation, which we can solve by rearranging and factoring:

x^2 + 5x - 72 = 0

Using calculator the solution is x = 6.35 or -11.35

Taking the positive value we have that x = 6.35

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There will be 4 plants left over after putting them in 7 equal rows.

To find out how many plants will be left over after putting them in 7 equal rows, we need to divide the total number of plants (438) by the number of rows (7) and then find the remainder.

We can use integer division to find out how many plants will be in each row:

438 ÷ 7 = 62

So each row will have 62 plants.

To find out how many plants will be left over, we can subtract the total number of plants from the product of the number of rows and the number of plants in each row:

438 - (7 x 62) = 438 - 434 = 4

Therefore, the number of plants remaining is 4 plants

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Answer:

m∠A = 50°

m∠B = 40°

AB = 15.5 units

Step-by-step explanation:

This is a right trangle, so we can use the Pythagorean Theorem to calculate AB.

AC² + BC² = AB²

AB² = 11.9² + 10²

AB² = 141.61 + 100

AB = √241.61

AB ≈ 15.5 units

Since this is a right triangle, we only need the angle measurement of either m∠A or m∠B to calculate the angle measurement of the other.

I will use sine function to calculate the angle measurements of A and B. Try to use a more precise value for AB than the rounded answer so our angle measurement will be closer to the actual angle.

sin(m∠B) = 10/15.54381

m∠B = sin⁻¹(10/15.54381)

m∠B = 40.04°
Round to nearest tenth, so m∠B = 40°

m∠A + m∠B + 90° = 180°

m∠A + 130° = 180°

m∠A = 50°

Answer:

[tex] - x - 7[/tex]

Step-by-step explanation:

[tex](4x - 6) - (5x + 1)[/tex]

A minus before the parentheses changes the signs in the parentheses to the opposite:

[tex] 4x - 6 - 5x - 1[/tex]

Subtract the number with x's seperately from the plain numbers:

[tex]4x - 5x - 1 - 6[/tex]

[tex] - x - 7[/tex]

we  should invest $479.17 each month to accumulate $81,000 in 12 years with an APR of 4 percent.

what is percent?

Percent is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin phrase "per centum," which means "by the hundred."

In the given question,

To calculate the monthly deposit needed to accumulate $81,000 in 12 years with an APR of 4%, we can use the formula for future value of an annuity:

FV = PMT x [(1 + r)ⁿ - 1] / r

Where:

FV = Future value of the investment (in this case, $81,000)

PMT = Monthly deposit

r = APR / 12 (monthly interest rate)

n = Number of months (12 years x 12 months per year = 144 months)

Substituting the given values, we get:

$81,000 = PMT x [(1 + 0.04/12)¹⁴⁴ - 1] / (0.04/12)

Simplifying this equation, we get:

PMT = $81,000 x (0.04/12) / [(1 + 0.04/12)¹⁴⁴ - 1]

PMT = $479.17

Therefore, you should invest $479.17 each month to accumulate $81,000 in 12 years with an APR of 4%.

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Thus, the total earning of Margot during summer is found to be $500.

Although the usage of percent and percentage differs slightly, they both signify the same thing. It is customary to use percent or the symbol (%) along with a numerical value.

Although the percentage word was not very ancient, the technique was often used. The Ancient Romans used to calculate fractions as multiples of 1/100 in the absence of the decimal system.

Given :

So,

12% of x =  60.00

x = 60.0 * 100 / 12

x = $500

Total earning amount = $500

Thus, the total earning of Margot during summer is found to be $500.

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Correct question:

Margot gave 12% of x number of dollars that she earned over the summer to an animal rescue shelter.

12% of x is about $60.00. Find the total earning of Margot during summer.

Answer:

-5, 0, 5

Step-by-step explanation:

Thus, the time taken by Colby to travel at each speed is-

5 hours at 12 mph and 6 hours at 10 mph.

In the study of physics, the three key variables are speed, distance, and time. How quickly or slowly an object moves through one point to another is determined by the concept of speed. The distance travelled in a unit of time is referred to as speed. Distance is exactly proportional to velocity and speed, while time is constant.

The formula for speed:

speed = distance/time

Then ,

time = distance / speed

Let speed S1 = 12 mph , for this time taken t1.

Let the speed S2 = 10 mph , for this time taken t2.

Since, in each distance is same say 'd'.

Total time t = 11 hours.

t = t1 + t2

t = d/s1  + d/s2

11 = d(1/12  +  1/10)

11 = d(12+10) / (12*10)

d = 11*120 / 22

d = 60 miles.

t1 = 60/12 = 5 hours

t2 = 60/10 = 6 hours.

Thus, the time taken by Colby to travel at each speed is-

5 hours at 12 mph

6 hours at 10 mph

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Answer:

the right answer is the third choice (-203)

Answer:

36 sq. inches assuming the shapes are same as well as sides

Answer: To write 93,000,000 in scientific notation, we need to move the decimal point to the left so that there is one non-zero digit to the left of the decimal point.

Counting the number of places we move the decimal point, we get:

93,000,000 = 9.3 x 10^7

Therefore, 93 million miles can be written in scientific notation as 9.3 x 10^7 miles.

Step-by-step explanation:

The values are; x = 10 and y = 10.67 according to the figure below.

Triangles with the same shape but different sizes are called similar triangles. In comparative triangles, relating points are harmonious and it are corresponding to compare sides.

Apply Pythagoras theorem for the small right angle triangle ΔBCE;

BC² = BE² + EC²

⇒ x² = 8² + 6²

⇒ x² = 64 + 36

⇒ x² = 100

Then, x = 10

Again apply Pythagoras theorem for upper right angle triangle ΔABE;

AB² = BE² + AE²

⇒ AB² = 8² + y²

⇒ AB² = 64 + y²

And also apply Pythagoras theorem for big right angle triangle ΔABC;

AC² = AB² + BC²

⇒ (6 + y)² = AB² + x²                      (put all above values)

⇒ (6 + y)² = 64 + y² + 10²    

⇒ 36 + y² + 12y = 64 + y² + 100

⇒ 12y = 164-36 = 128

⇒ y = 10.67

Therefore, the values are; x = 10 and y = 10.67

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The values are; x = 10 and y = 10.67 according to the given figure.

If the hypotenuse and one of the legs of one right triangle are proportional to their relative lengths in the second right triangle, then the two triangles are comparable.

Right triangles don't all look the same. There are two right triangles that we can have where this is not the case. For two triangles to be comparable, the ratios comparing the lengths of their respective sides must all be equal.

Use the Pythagorean theorem to solve the tiny right triangle in ΔBCE;

BC² = BE² + EC²

x² = 8² + 6²

x² = 64 + 36

x² = 100

Then, x = 10

Apply the Pythagorean theorem once more to the triangle with upper right angle ΔABE;

AB² = BE² + AE²

AB² = 8² + y²

AB² = 64 + y²

Apply the Pythagorean theorem to the large right triangle ΔABC as well.

AC² = AB² + BC²

(6 + y)² = AB² + x² (put all above values)

(6 + y)² = 64 + y² + 10²    

36 + y² + 12y = 64 + y² + 100

12y = 164-36 = 128

y = 10.67

The values are:

x = 10 and y = 10.67

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With the given values of circle, the value of x=3 and VS = 5.

What is circle?

A circle is a geometrical shape consisting of all the points in a plane that are equidistant from a given point called the center of the circle.

Since VZ = UZ, then Z is on the perpendicular bisector of QR and ST. This means that Z divides QR and ST into two equal parts.

Using the midpoint formula, we can find the midpoint of QR and ST:

Midpoint of QR = ( (1/2)*(4x-2) , 0 ) = (2x-1, 0)

Midpoint of ST = ( (1/2)*10 , 8 ) = (5, 8)

Since Z is the midpoint of QR and ST, we can set up two equations:

2x-1 = 5

0 = 8

The second equation is not possible, so we ignore it. Solving the first equation gives us:

2x-1 = 5

2x = 6

x = 3

Now that we know x, we can find the length of VS:

VS = ST - TV = ST- (ST/2) = 10 - (5) = 5

Therefore, x = 3 and VS = 5.

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A schedule of cash receipts for the months of August and September, all amounts rounded to the nearest dollar has been attached.

The balance of money owed to a business for goods or services delivered or utilised but not yet paid for by clients is known as accounts receivable (AR). On the balance sheet, accounts receivable are shown as a current asset. Any sum of money that clients owe for purchases they made using credit is known as AR.

The total collection for the months has been provided:

May = $14100

June = $48025

July = $50537

August = $46623

September = $58105

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Answer:

the coordinate is (11,1)

The percentage rate of change per day is approximately 0.1932%.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

To represent the number of users on the website after t years, we can use the exponential growth formula:

[tex]N_t = N_0 \times e^{rt}[/tex]

where [tex]N_0[/tex] is the initial number of users, r is the growth rate, and t is the time in years.

Given that [tex]N_0[/tex] = 6300 and the growth rate is 73%, or 0.73, per year,

The function is:

[tex]N_t = 6300 \times e^{0.73t}[/tex]

To find the percentage rate of change per day, we can use the formula:

[tex]r = (e^{0.73/365} - 1) \times 100%[/tex]

where 0.73/365 is the daily growth rate.

Solving for r, we get:

[tex]r = (e^{0.73/365} - 1)[/tex] x 100%

r ≈ 0.1932%

Therefore,

The percentage rate of change per day is approximately 0.1932%.

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In the given equation the value of mº is 17°. Answer: (D) 49 = 32° + mº.

A triangle is a geometrical figure with three straight sides and three angles. It is the simplest polygon and the building block for more complex shapes.

There are several kinds of triangles:

Equilateral triangle: This is a triangle with three equal sides and three equal angles of 60 degrees each.

Isosceles triangle: This is a triangle with two equal sides and two equal angles.

Scalene triangle: This is a triangle with no equal sides or angles.

Right-angled triangle: This is a triangle with one angle of 90 degrees, which is also known as the right angle. The side opposite to the right angle is the longest side, called the hypotenuse, while the other two sides are called legs.

Acute triangle: This is a triangle in which all three angles are acute angles, meaning they are less than 90 degrees.

Obtuse triangle: This is a triangle in which one angle is an obtuse angle, meaning it is greater than 90 degrees. The other two angles are acute angles.

The equation that can be used to solve for mº depends on the given information or conditions. If the problem provides the measures of two angles and their sum, then we can use the equation:

A + B + m = 180°

where A and B are the measures of the given angles, and m is the measure of the unknown angle in degrees.

However, if the problem only gives the measures of two angles and their sum is not required, we can use the equation:

A + B = m

where A and B are the measures of the given angles, and m is the measure of the unknown angle in degrees.

Based on the given information in the question, we are only provided with the measures of two angles, 49° and 32°, and there is no indication that their sum is required. Therefore, the correct equation to solve for mº is:

49° = 32° + mº

We can simplify this equation to solve for mº:

49° - 32° = mº

mº = 17°

Therefore, the value of mº is 17°. Answer: (D) 49 = 32° + mº.

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Answer:

simplifies to 5[tex]\sqrt{2}[/tex]

Step-by-step explanation:

using the rule of radicals

[tex]\sqrt{ab}[/tex] = [tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex]

then

[tex]\sqrt{32}[/tex] = [tex]\sqrt{16(2)}[/tex] = [tex]\sqrt{16}[/tex] × [tex]\sqrt{2[/tex] = 4[tex]\sqrt{2}[/tex]

then

[tex]\sqrt{2}[/tex] + [tex]\sqrt{32}[/tex]

= [tex]\sqrt{2}[/tex] + 4[tex]\sqrt{2}[/tex] ( the 2 terms have like radicals so add coefficients )

= 5[tex]\sqrt{2}[/tex]

In the tangent line , Value of [tex]m\angle{VKT}[/tex] is A)54°.

A circle's tangent is a single-pοinted line that intersects the circle. The "pοint οf tangency" is the lοcatiοn where the circle and the tangent crοss. The radius οf the circle, with which it crοsses, is perpendicular tο the tangent. All curved shapes can be thοught οf as tangent. As a tangent is a line, it has an equatiοn as well.

The line that οnly tοuches the circumference οf a circle οnce is said tο be tangent tο it. Fοr each circled pοint, there can οnly be οne tangent. The tangent's pοint οf tangency is where it jοins the circle.

Here the given circle arc length mVPT = 234°. Then

arc length οf mVT = 360°-234° = 126°

Nοw using tangent line fοrmula in circle then,

=> [tex]m\angle{VKT}=\frac{mVPT-mVT}{2}[/tex]

=> [tex]m\angle{VKT} =\frac{234\textdegree-126\textdegree}{2}[/tex]

=> [tex]m\angle{VKT} =\frac{108\textdegree}{2} =54\textdegree[/tex].

Hence the correct option is A)54°.

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Answer:

1966

Step-by-step explanation:

910x2=1820

23x2= 46

1820=46=1966

The expression-4 gives the total number of spaces traveled during the 4 turns can be found using integers or its absolute values.

What are integers?

Integers are the collection of negative, positive numbers including zero. The positive integers lie on the right side of the zero on the number line and the negative lie on the left side of zero on number line. The positive numbers are larger than the negative numbers. Also zero is greater than all negative numbers.The absolute value/modulus of any integer is its magnitude without sign. It is the distance of any given integer from zero.

Here the spaces travelled in each turn is given as +7, +2, +5 and -8.

The total sum of the spaces=(+7) + (+2) + (+5) + (-8)

                                             = 7+2+5-8

                                             =6

The expression in option 1: |7 + -2 + 5 + - 8| which is not equal to 6

The expression in option 2: 7 - 2 + 5 - 8 which is not equal to 6

The expression in option 3: |(+7) + (-2) + (+5) + (- 8)| which is not equal to 6

The expression in option 4: |7 | + |-2| + |5| + |- 8| which is equal to 6

∴The correct expression is option-4

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the measure of the angle formed by the beam and the ground is approximately 57.76 degrees. the measure of the angle formed by the beam and the ground is approximately  [tex]57.76^\circ[/tex] degrees.

What is Right triangles?

A right triangle is a type of triangle that has one angle measuring 90 degrees (a right angle). The side opposite to the right angle is called the hypotenuse, and the other two sides are called the legs.

The Pythagorean theorem is a fundamental concept in mathematics that relates to right triangles, stating that the square of the length of the hypotenuse is equal to the sum of the squares of  lengths of  two legs.

To solve this problem, we can use trigonometry and the properties of right triangles.

First, let's draw the picture:

In this diagram, we have a right triangle formed by the beam, the wall, and the ground. The height of the triangle is given as 16 feet, and the length of the hypotenuse (the beam) is 30 feet.

We need to find the length of the base of the triangle (the distance between the wall and the bottom of the beam), which we can call d.

Using the Pythagorean theorem, we can find the length of the opposite side of the triangle:

length = √(beam² - wall²)

length = √(30² - 16²)

length = √(900 - 256)

length = √644

length ≈ 25.37 feet

Now we can use the tangent function to find the angle θ:

tan(θ) = opposite/adjacent

tan(θ) = ground/wall

tan(θ) = 25.37/16

tan(θ) ≈ 1.586

To find θ, we need to take the inverse tangent (or arctangent) of 1.586:

[tex]\Theta = tan-1(1.586)[/tex]

 [tex]\Theta \approx 57.76^\circ[/tex] degrees

Therefore, the measure of the angle formed by the beam and the ground is approximately  [tex]57.76^\circ[/tex] degrees.

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Answer:

Step-by-step explanation:

Answer:

The function f(x) = x represents a straight line with a slope of 1, passing through the origin (0,0).

If we consider the interval from x = 2 to x = 9, we can see that the line starts at f(2) = 2 and ends at f(9) = 9.

Therefore, over the interval from x=2 to x=9, the function f(x) = x increases in a straight line fashion from 2 to 9.

In other words, the slope of the line is positive, and the function is monotonically increasing on this interval.

After analyzing the graph and using the percentages, it can be inferred that Streetcar and Cable Car are the preferred transportation for 168 residents.

To draw conclusions from the graph, we can use the percentages to calculate the number of residents who prefer each transportation method.

Number of residents who prefer Walk: 40% of 400 = 0.4 x 400 = 160

Number of residents who prefer Bicycle: 8% of 400 = 0.08 x 400 = 32

Number of residents who prefer Streetcar: 15% of 400 = 0.15 x 400 = 60

Number of residents who prefer Bus: 10% of 400 = 0.1 x 400 = 40

Number of residents who prefer Cable Car: 27% of 400 = 0.27 x 400 = 108

From the given circle graph, we can see that the preferred transportation methods and their percentages are:

Walk: 40%

Bicycle: 8%

Streetcar: 15%

Bus: 10%

Cable Car: 27%

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[tex]f(x) = -5 * 2[/tex] is one conceivable exponential function (-x). The coordinates of this function are (0, -5) and (-2, -20).

Because the exponent's base falls between 0 and 1, this function depicts exponential decay. The output of the function decreases at an increasing rate as x rises because the value of 2(-x) shrinks. The function starts at a number larger than zero and lowers with time, as shown by the negative sign. The initial value of -5 denotes the function's beginning point. As a result, when x rises, the function's output gets closer to zero but never quite hits it.

Because to the diminishing nature of the exponent's base, the function [tex]f(x) = -5 * 2(-x)[/tex]  generally indicates exponential decay.

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