find the missing parts using trigonometry functions . round to the nearest tenth

Answer:

even

Step-by-step explanation:

It lands on 10, and 8, take that, divide by 2, and mutiply by 8.

Answer:

an = 6 * 2^(n-1)

Step-by-step explanation:

To find the general term of the sequence 6, 12, 24, 48, ... , we need to observe that each term is obtained by multiplying the previous term by 2. Therefore, the sequence is a geometric sequence with first term a = 6 and common ratio r = 2.

The formula for the nth term of a geometric sequence is:

an = a * r^(n-1)

Substituting a = 6 and r = 2, we get:

an = 6 * 2^(n-1)

Therefore, the general term of the sequence 6, 12, 24, 48, ... is given by the formula an = 6 * 2^(n-1).

Based on the information provided, the total tax this person will need to pay is $2,622,2.

The amount of money you pay for taxes depends on your income as there are specific percentages to be paid. In the case the income is $25000, this income matches the range $7,825 - $31,850, and therefore you should pay 50 plus 15 percent of the amount over 7,825. Based on this, let's calculate the money to be paid:

$25,000 - $7852 = $17, 148 x 0.15 (15%) = $2,572.2 + $50 = $2622.2

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The area of the circle is 200.96 square kilometers, the area of the polygon is 181.02 square kilometers and the shaded region is 19.94 square kilometers.

The area A of a circle with radius r can be found using the formula:

A = πr^2

Plugging in the given value of radius, we get:

A = π(8 km)^2

Simplifying the expression, we get:

A = 64π km^2

So, we have

A ≈ 64 x 3.14 km^2

A ≈ 200.96 km^2

To find the area of an octagon with a radius of 8 km, we can break it up into eight congruent isosceles triangles.

So, we have

Area = 8 * Area of triangle

This gives

Area = 8 * 1/2 * r² sin(central angle)

So, we have

Area = 8 * 1/2 * 8² sin(45)

Evaluate

Area = 181.02

The shaded area is

Shaded = 200.96 - 181.02

Shaded = 19.94

Hence, the shaded area is 19.94 square kilometers.

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The equation for a cosecant function with vertical asymptotes found at x = nπ such that n is an integer is `y = csc(x - nπ)`.

The cosecant function is the reciprocal of the sine function. The cosecant of a particular angle in a right triangle is the ratio of the hypotenuse to the side opposite the angle.

The cosecant function is written as follows: `csc(x) = 1/sin(x)`

Asymptotes are straight lines or curves that a graph approaches but never touches.

As a result, the graph of a curve gets closer and closer to the asymptote as it moves further away from the origin, but it never touches it.

For a function to have vertical asymptotes at `x = nπ`, the denominator of the function should equal zero. In the cosecant function, the denominator is equal to `sin(x - nπ)`.

Therefore, the vertical asymptotes are where `sin(x - nπ) = 0`.

The general form of the cosecant function is `y = A csc(B(x - C)) + D`.

The constants A, B, C, and D are used to alter the form of the graph of the cosecant function.

They can be used to alter the amplitude, period, phase shift, and vertical displacement of the graph, respectively.

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According to the information in the table, the bus arrives in Newtown at 11:09. So this trip would take 31 minutes.

To find out what time the bus arrives in Newtown we must analyze the table. In this case we can see row 3 that has the information about the bus route that leaves at 11:38.

The fourth city that this bus visits is Newtown, where it arrives at 11:09 after stopping in Milton and Leek. So the arrival time in Newtown would be 11:09.

On the other hand, to find how long the trip took we must compare both times (10:38, 11:09) and find the difference. So, for it to be 11:00, there would be 22 minutes left. Then add 9 minutes, so the total would be 31.

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Part A: If assuming 0 means the white part, then nο.

Part B: If assuming 0 means the black part, then yes.

The prοbability οf an οccurrence is a figure that represents hοw likely it is that the event will take place. In terms οf percentage nοtatiοn, it is expressed as a number between 0 and 1, οr between 0% and 100%. The higher the likelihοοd, the mοre likely it is that the event will take place.

Part A: If assuming 0 means the white part, then nο, there is a chance tο hit the black circle but the circle is οnly abοut 1/9th, then the whοle target, sο putting it in numbers, it wοuld be abοut 1/9 cοmpared tο 8/9.

8/ 9 has a higher chance.

Part B: If assuming 0 means the black part, then yes, as I said in Part A, it's basically cοmparing 1/9 tο 8/9.

1/9 being the black part and 8/9 being the white [tex]8/9 > 1/9[/tex] which means the white pοrtiοn has a higher prοbability.

Hence, Part A: If assuming 0 means the white part, then nο.

Part B: If assuming 0 means the black part, then yes.

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let's call our number hmmm Z, now, let's reduce it by 17%, so 100% - 17% = 83%, so the new size for Z is 83% off the original, hmm how much is that?

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{83\% of Z}}{\left( \cfrac{83}{100} \right)Z}\implies 0.83Z[/tex]

now, let's increase it by 12%, so the new size will be 100% + 12% = 112%, so 112% of 0.83Z, let's check how much is that

[tex]\stackrel{\textit{112\% of 0.83Z}}{\left( \cfrac{112}{100} \right)0.83Z}\implies 0.9296Z[/tex]

now, let's convert that to a percent format by simply multiplying it by 100, so that'd be 100 * 0.9296Z = 92.96% of Z.  Well, hell 92.96% is less than 100%, so is really 100% - 92.96% = 7.04% less.

So, instead of all that rigamarole, we could have just reduced Z by 7.04% in one fell swoop and obtain the same thing.

The value of y is 7.5 for the given similar triangle.

The sides of comparable triangles are relative to one another. For instance, if the proportion of the lengths of two comparing sides of one triangle to another is 2:1, then the other two relating sides will likewise be in the proportion of 2:1.

According to the figure; two triangles are similar because its identities,

So, (2y + 25) = 2 times of (2y + 5)

we can write as,

⇒ (2y + 25) = 2 × (2y + 5)

⇒ 2y + 25 = 4y + 10

⇒ 2y - 4y = 10 - 25

⇒ - 2y = - 15

⇒ y = 15/2 = 7.5

Therefore, the value of y is 7.5 of the given similar triangle.

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The acceleration function is a(t) = 6(t - 3).

Acceleration is the rate at which an object changes its velocity. The acceleration function is a mathematical function that gives the acceleration of an object at any given time

To find the acceleration function, we need to take the second derivative of the given velocity function f(t).

f(t) = (t - 3)^3 + t

First, let's find the first derivative of f(t)

f'(t) = 3 × (t - 3)^2 + 1

Now, let's find the second derivative of f(t) ( the acceleration function )

a(t) = f''( t )

= d/dt [3 × (t - 3)^2 + 1]

Do the derivation

= 6(t - 3)

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The given question is incomplete, the complete question is:

The velocity (in meters per second) with which a roller coaster moves can be given by the function f(t) = (t - 3)^3 + t. find the acceleration function a(t).

value of variable x on the parallel line is 12.

In geometry, a co exterior angle is an angle that is formed outside of a geometric figure by extending one of its sides. More specifically, a co exterior angle is formed when a transversal intersects two parallel lines, and it is an angle that is located outside of the parallel lines and on the same side of the transversal as one of the angles formed by the intersection.

Given

m ║n

8x+6 and 9x-30 are co exterior angle

8x+6+9x-30=180°

17x-24=180°

x=12

hence, value of variable x on the parallel line is 12.

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In conclusion the area of the segment is approximately 32.76 square units.

How to find?

To find the area of a segment of a circle, we can use the formula:

Area of segment = (θ/360)πr²2 - (1/2)rsin(θ)

where:

θ is the central angle of the segment in degrees

r is the radius of the circle

π is a constant approximately equal to 3.14

sin is the sine function

In this case, the central angle is 50° and the radius is 15. Substituting these values into the formula, we get:

Area of segment = (50/360)π(15)²2 - (1/2)(15)sin(50)

= (5/36)π(225) - (1/2)(15)(0.766)

= 32.76 square units (rounded to two decimal places)

Therefore, the area of the segment is approximately 32.76 square units.

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The distributive property states that we can distribute a factor across a sum or difference, add or subtract like terms by adding or subtracting their coefficients and The power rule states that when raising a power to another power, we multiply the exponents. and Algebraic Properties explained as

Journal Entry 1:

Today, I learned about the algebraic properties of addition and multiplication. These properties are commutative, associative, and distributive. The commutative property states that the order in which we add or multiply numbers does not affect the result. For example, 2+3 is the same as 3+2, and 2x3 is the same as 3x2. The associative property states that we can group numbers in different ways without changing the result. For example, (2+3)+4 is the same as 2+(3+4), and (2x3)x4 is the same as 2x(3x4). The distributive property states that we can distribute a factor across a sum or difference. For example, 2x(3+4) is the same as 2x3 + 2x4.

Journal Entry 2:

Today, I learned about algebraic expressions and how to simplify them using the properties of addition and multiplication. An algebraic expression is a combination of numbers, variables, and operations. For example, 2x + 3y - 4z is an algebraic expression. To simplify an expression, we use the properties of addition and multiplication to combine like terms and simplify the expression as much as possible. Like terms are terms that have the same variables raised to the same powers. For example, 2x and 5x are like terms, but 2x and 5y are not. We can add or subtract like terms by adding or subtracting their coefficients. For example, 2x + 5x is 7x. We can also multiply terms by using the distributive property. For example, 2(3x + 4y) is 6x + 8y.

Journal Entry 3:

Today, I learned about algebraic expressions with exponents. An exponent is a small number written to the right of a base number that indicates how many times to multiply the base by itself. For example, in 2³, the base is 2 and the exponent is 3. To simplify an expression with exponents, we use the properties of exponents, such as the product rule and the power rule. The product rule states that when multiplying two powers with the same base, we add their exponents. For example, 2³ x 2² is 2^(3+2) or 2^5. The power rule states that when raising a power to another power, we multiply the exponents. For example, (2²)³ is 2^(2x3) or 2^6. We can also simplify expressions with exponents by combining like terms, just like we did with expressions without exponents.

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1.2.4Journal: Algebraic Properties and ExpressionsJournalAlgebra I Sem 1Name:Date:Scenario:The ArcadeInstructions:

•View the video found on page 1 of this journal activity

.•Using the information provided in the video, answer the questions below

.•Show your work for all calculation

Answer:

∡7 = 70°

Step-by-step explanation:

∡7 and ∡8 are supplementary angles

∡7 + ∡8 = 180°

If:

∡8 = 110°

Then:

∡7 = 180 - 110

∡7 = 70°

The length of the line segment between C(-3,3) and D(3,-3) is 6√2 units.

Given that

C(-3, 3) and D(3, -3)

To find the length of the line segment between points C(-3,3) and D(3,-3), we can use the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

Plugging in the coordinates for C(-3,3) and D(3,-3), we get:

d = √((3 - (-3))^2 + (-3 - 3)^2)

Evaluate

d = √(6^2 + (-6)^2)

d = √72

Simplifying the square root, we can write the length as:

d = 6√(2)

Therefore, the length of the segment CD is 6√2 units.

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In the given parallelogram vwxy: x = 3; m∠Y = 65° and ∠YVX = 61° by (alternate interior angle).

A quadrilateral with the opposing sides parallel is called a parallelogram (and thus opposite angles equal).

In the given parallelogram vwxy:

Parallel sides are equal.

So, 2x = 6

x = 6/2 = 3

Opposite pair of angles are also equal.

So,

∠W = ∠Y = 65°

Now,

∠YVX = 61° (alternate interior angle).

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On the first day, the members earned 90% of their daily goal of $50.00 which is equal to $45.00 (0.9 x $50.00 = $45.00).

On the second day, the members earned 6% more than their daily goal of $50.00 which is equal to $53.00 ($50.00 + 0.06 x $50.00 = $53.00).

On the third day, the members earned 14% less than their daily goal of $50.00 which is equal to $43.00 ($50.00 - 0.14 x $50.00 = $43.00).

Therefore, the members earned a total of $45.00 + $53.00 + $43.00 = $141.00 from ticket sales on all three days.

The members of the school club earned $141.00 from ticket sales over the three days.

percentage, a relative value indicating hundredth parts of any quantity.

To find out how much money the members of the school club earned from ticket sales on all three days

Let us calculate the amount of money earned each day and then add them together.

On the first day, the members earned 90% of their daily goal, which is:

0.9 x $50.00 = $45.00

On the second day, they earned 6% more than their daily goal, which is:

$50.00 + 0.06 x $50.00 = $53.00

On the third day, they earned 14% less than their daily goal, which is:

$50.00 - 0.14 x $50.00 = $43.00

To find the total amount of money earned over the three days, we simply add the amounts earned on each day:

$45.00 + $53.00 + $43.00 = $141.00

Therefore, the members of the school club earned $141.00 from ticket sales over the three days.

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(1)  The scale factor for this dilation is 1/6.

(2) The scale factor for this dilation is 6.

(3) False. A dilation with a scale factor less than 1 is a reduction, not an enlargement.

1. To find the scale factor for the dilation with center at the origin that maps point M(-12,18) to M(-2,3) we can use the formula:

new coordinate / old coordinate = scale factor

For the x-coordinate:

-2 / -12 = 1/6

For the y-coordinate:

3 / 18 = 1/6

Since both coordinates have the same scale factor, we can conclude that the scale factor for this dilation is 1/6.

2. To find the scale factor for the dilation with center at the origin that maps point M(-2,3) to M'(-12,18) we can again use the formula:

new coordinate / old coordinate = scale factor

For the x-coordinate:

-12 / -2 = 6

For the y-coordinate:

18 / 3 = 6

Since both coordinates have the same scale factor, we can conclude that the scale factor for this dilation is 6.

3. False. A dilation with a scale factor less than 1 is a reduction, not an enlargement. An enlargement is a dilation with a scale factor greater than 1.

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We find sets of three positive single digit integers with a mean of 4 and a range of 5 is {1, 5, 6} and another possible set is {2, 4, 6}.

Here mmr is mean, median, and range of a set of numbers. The mean of a set of numbers is the sum of all the numbers divided by the total number of numbers in the set. The median of a set of numbers is the middle number when the numbers are arranged in order.

The range of a set of numbers is the difference between the largest and smallest numbers in the set.

To find two sets of three positive single digit integers with a mean of 4 and a range of 5, we can use the formula:

Mean = (sum of numbers) / (number of numbers)

Since we want the mean to be 4, the sum of the three numbers in each set must be 12. Since we want the range to be 5, the largest number in each set must be 5 more than the smallest number.

One possible set of three numbers that satisfies these conditions is {1, 5, 6}. The mean of this set is (1+5+6)/3 = 4, and the range is 6-1 = 5.

Another possible set of three numbers that satisfies these conditions is {2, 4, 6}. The mean of this set is (2+4+6)/3 = 4, and the range is 6-2 = 4.

Therefore, two sets of three positive single digit integers with a mean of 4 and a range of 5 are {1, 5, 6} and {2, 4, 6}.

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Answer: The first graph

Step-by-step explanation: because the vertex is (3,-1)

Then plug in number like is 1 is x solve for y which will be -3

if x is 2, y is -3/2

x is 3, y is -1

x is 4, y is -3/2

x is 5, y is -3

Answer:

Vertex at (3,-1) and flipped upside down (opening downward)

*looks like the first one of the first attachment*

Step-by-step explanation:

The normal untransformed parabola is a U-ish, V-ish curve that has its vertex at (0,0) and it opens upward. There are "nice" points at (1,1) and (2,4) also at (-1,1) and (-2,4) Your parabola has had four changes made to it:

□The negative sign in front on your equation will FLIP the whole thing upside down so it opens downward.

□The 1/2 in front will "smash" the parabola so it is flatter and wider.

□The - 3 that's in close next to the x will shift (slide, translate) the whole thing to the right 3 units **left and right shifts are kind of the opposite of what you might think: - shifts right and + shifts left**

□The -1 tacked on to the end will shift (slide, translate) the whole thing down one unit. **more intuitive, - shifts down and + shifts up**

the number of students who take only Physics is 15.

An equation is a mathematical statement that shows that two expressions are equal. It is typically written using an equal sign (=) between the two expressions. The expressions on either side of the equal sign are called the left-hand side (LHS) and the right-hand side (RHS) of the equation.

Let's denote the number of students who take only Physics by x.

We can use these equations to solve for x, the number of students who take only Physics:

Substituting the expression for the number of Physics students in terms of the number who take both, we get:

20 = number who take both + 3(number who take both)

20 = 4(number who take both)

number who take both = 5

this value in the expression for the number of Physics students, we get:

number of Physics students = 3(number who take both) = 15

Finally, we can use the first equation to solve for the number of students who take only Physics:

40 = 15 + 2(number of students who take only Physics) - 5

40 = 10 + 2(number of students who take only Physics)

number of students who take only Physics = 15

Therefore, the number of students who take only Physics is 15.

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Step-By-Step:

The height triangle has a height of 17.3 yards and a width of 12 yards. 3. Finally, connect the two triangles to create the pyramid's surface area. The surface area of the pyramid is 116.7 yards².

Answer:

116.7 yards².

Answer:

The first step is to find the area of the base triangle:

Area = (1/2) x base x height

Area = (1/2) x 20 yards x 17.3 yards

Area = 173 square yards

To find the surface area of the pyramid, we need to find the areas of the four triangular faces and add them to the base area.

Area of a triangular face = (1/2) x base x height

Area of each face = (1/2) x 20 yards x 12 yards

Area of each face = 120 square yards

Now we can add up all the areas:

Surface area = base area + area of four faces

Surface area = 173 square yards + 4 x 120 square yards

Surface area = 653 square yards

Therefore, the surface area of the pyramid is 653 square yards.

b. i) The probability of winning all three matches is 1/27.

ii) The probability of winning more times than losing is 3/27.

iii) 7/27

iv) 2/27

c. many factors that can influence the outcome of a match such as the skill of the players, the tactics used by the teams and the condition of the playing surface.

Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

a. Tree diagram:

Match 1   | Match 2   | Match 3

--------------------------------

Win       | Win       | Win

Draw      | Draw      | Draw

Lose      | Lose      | Lose

b. i) The probability of winning all three matches is P(Win, Win, Win) = (1/3) * (1/3) * (1/3) = 1/27

ii) The probability of winning more times than losing is

P(Win, Win, Draw) + P(Win, Draw, Win) + P(Draw, Win, Win)

= (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3)

= 3/27

iii) The probability of losing at least one match is

P(Lose, Win, Win) + P(Win, Lose, Win) + P(Win, Win, Lose) + P(Lose, Lose, Win) + P(Lose, Win, Lose) + P(Win, Lose, Lose) + P(Lose, Lose, Lose)

= (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3)

= 7/27

iv) The probability of either drawing or losing all three matches is

P(Draw, Draw, Draw) + P(Lose, Lose, Lose)

= (1/3) * (1/3) * (1/3) + (1/3) * (1/3) * (1/3)

= 2/27

c. It is not very realistic to assume that the outcomes are equally likely in this case because there are many factors that can influence the outcome of a match such as the skill of the players, the tactics used by the teams and the condition of the playing surface.

Therefore, it is unlikely that all outcomes are equally likely as assumed in this case.

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After answering the presented question, we can conclude that equation 0.1492063492063492 * 100% Percentage increase = 14.9% As a result, the average yearly pay has increased by 14.9%.

An equation in mathematics is a statement that states the equality of two expressions. An equation is made up of two sides that are separated by an algebraic equation (=). For example, the argument "2x + 3 = 9" asserts that the phrase "2x + 3" equals the number "9". The purpose of equation solving is to determine the value or values of the variable(s) that will allow the equation to be true. Equations can be simple or complicated, regular or nonlinear, and include one or more elements. The variable x is raised to the second power in the equation "x2 + 2x - 3 = 0." Lines are utilised in many different areas of mathematics, such as algebra, calculus, and geometry.

To determine the % increase in mean annual wage, find the difference between the new and old salaries, divide it by the old salary, and multiply by 100.

(new salary - previous salary) / old salary * 100% = percentage increase

(28960 - 25200) / 25200 * 100% Percentage increase = 0.1492063492063492 * 100% Percentage increase = 14.9%

As a result, the average yearly pay has increased by 14.9%.

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Answer:

The height is 9.73 cm.

Step-by-step explanation:

We can start by finding the radius of the cylinder:

r = d/2 = 4/2 = 2 cm

Then, we can use the formula for the surface area of a cylinder to solve for the height:

SA = 2πrh + 2πr²

130 = 2π(2)(h) + 2π(2)²

130 = 4πh + 8π

122 = 4πh

h ≈ 9.73 cm (rounded to the nearest hundredth)

So the height of the can is approximately 9.73 cm.

the runner still needs to run a distance of 15 km to finish the marathon.

To find out how much distance remains to be run in the marathon, we can first calculate how far the runner has already run in the last 3 hours. We can use the formula:

distance = speed x time

where speed is in meters per second and time is in hours.

The runner has been running at a constant speed of 5/3 m/s for 3 hours, so the distance covered so far is:

distance = (5/3) m/s x 3 hours x 3600 seconds/hour

distance = 6000 meters or 6 km

Therefore, the runner has already completed 6 km of the marathon, which means that the remaining distance to be run is:

remaining distance = total distance - distance already covered

The total distance of the marathon is 21 km, so the remaining distance is:

remaining distance = 21 km - 6 km

remaining distance = 15 km

Therefore, the runner still needs to run 15 km to finish the marathon.

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The x- and y-intercepts of the function f(x) = 2x² + x + 5 is A.

we put x as equal to 0 first and we get y = 5 (y=f(x))

once we get y = 5 we can replace it in the equation and we get 2x^2+x=0 solving the quadratic equation, we get x=0, -1/2

The quadratic equation can be solved by taking x common then we are left with (2x+1).

the x we take common is equal to 0

(2x+1) = 0 gives x=-1/2

hence the x intercept = -1/2, 0 and y intercept =5

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Answer:

24

Step-by-step explanation:

Answer: 24 hats

Step-by-step explanation:

1. Find common multiples: 2: 2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36   6: 6,12,18,24,30,36   8: 8,16,24,32

2. Find the least common multiple: 24

A can also be represented as A(2,-11π/6) and B can be represented as B(3,-π/6). Using the distance formula again, we have: distance = sqrt(2^2 + 3^2 - 223*cos(-11π/6 - (-π/6))) = sqrt(13)

What is trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles.

a) To find the distance between two points in polar coordinates, we can use the Pythagorean theorem to calculate the distance in the radial direction and then use the law of cosines to take into account the angular separation. Thus,

distance = sqrt((r2-r1)^2 + (r1^2 + r2^2 - 2r1r2*cos(θ1 - θ2)))

Simplifying this expression, we get:

distance = sqrt(r1^2 + r2^2 - 2r1r2*cos(θ1 - θ2))

Using the identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we can rewrite the expression as:

distance = sqrt(r1^2 + r2^2 - 2r1r2cos(θ1)cos(θ2) - 2r1r2sin(θ1)sin(θ2))

Using the identity cos^2(x) + sin^2(x) = 1, we can simplify the expression further:

distance = sqrt(r1^2 + r2^2 - 2r1r2*cos(θ1-θ2))

b) If θ1 = θ2, then the points lie on the same radial line and are separated only by their radial distances from the origin. In this case, the distance formula simplifies to:

distance = sqrt((r2-r1)^2) = abs(r2-r1)

This is what we would expect, as the distance between two points on the same radial line is simply the absolute difference in their radial distances from the origin.

c) If θ1 - θ2 = 90 degrees, then the points are separated by the maximum angular distance and lie on perpendicular radial lines. In this case, the distance formula simplifies to:

distance = sqrt(r1^2 + r2^2)

This is also what we would expect, as the distance between two points on perpendicular radial lines is simply the Pythagorean sum of their radial distances from the origin.

d) Let's choose two points in polar coordinates: A(2,π/6) and B(3,5π/6). Using the distance formula derived in part (a), we have:

distance = sqrt(2^2 + 3^2 - 223*cos(5π/6 - π/6)) = sqrt(13)

Now let's express these same points in different polar representations.

Therefore, a can also be represented as A(2,-11π/6) and B can be represented as B(3,-π/6). Using the distance formula again, we have:

distance = sqrt(2^2 + 3^2 - 223*cos(-11π/6 - (-π/6))) = sqrt(13)

As expected, the distance between the two points is the same regardless of their polar representations.

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