Determine whether the function is linear. If it is, identify the rate of change. X -7, -5, -3, -1, 0 Y 11, 14, 17, 20, 23​

Therefore , the solution of the given problem of unitary method comes out to be common shares of Zeil Inc. is 2.23%.

This common convenience, already-existing variables, or all important elements from the original Diocesan adaptable study that followed a particular methodology can all be used to achieve the goal. Both of the crucial elements of a term affirmation outcome will surely be missed if it doesn't happen, but if it does, there will be another chance to get in touch with the entity.

Here,

Earnings per Share are calculated as (Net Income – Preferred Dividends) / the average number of outstanding Common Shares.

=>  Market price per share / earnings per share is the Price-Earnings Ratio.

=> Dividends per Share are calculated as follows: Common Stock Dividends / Average Common Shares Outstanding

=>  Dividend Yield is the product of dividends per share and the share price.

=>  (Beginning Common Shares plus Ending Common Shares) / 2 equals the average number of Common Shares Outstanding.

=>  Starting common shares equals ending common shares, which is

=>  $3,500,000 / $25, or 140,000.

(a) The earnings per share are ($424,000 - $0) / 140,000, which equals $3.03.

The ordinary stock price of Zeil Inc.

(b) The price-earnings ratio for Zeil Inc.'s common shares is 11.20 divided by 3.03, or 3.69.

(c) Dividends per Share: $35,000./140,000. = $0.25

Therefore, $0.25 in dividends are paid per unit of Zeil Inc. common stock.

(d) Dividend Yield: $0.25 divided by $11.20 equals 0.0223, or 2.23%.

The common shares of Zeil Inc. is 2.23%.

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The estimated volume of the coffee cup is approximately 152.8 cubic centimeters.

It is the perimeter of the circle, which can be found by multiplying the diameter of the circle by pi (π), a mathematical constant that is approximately equal to 3.14.

The volume of a right cylinder is:

V = πr²h

We are given the height of the coffee cup as h = 8.3 cm. To find the radius,

C = 2πr

We are given the circumference of the coffee cup as C = 14.9 cm. Solving for r, we have:

14.9 = 2πr

r = 14.9 / (2π) ≈ 2.372 cm

Now we can substitute these values into the formula for the volume of a cylinder:

V = πr²h

V = π(2.372)²(8.3)

V ≈ 152.8 cubic centimeters

Therefore, the estimated volume of the coffee cup is approximately 152.8 cubic centimeters.

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To remove the y-term in  the given linear equation in two Variable multiply the equation by 4 then add these two equations.

The equation for a linear equation in one variable is written as ax+b = 0, where a and b are two integers, and x is a variable. This equation has only one solution. For instance, the linear equation 2x+3=8 only has one variable. As a result, this equation has a single solution, x = 5/2. Yet, there are two solutions to a linear equation with two variables.

According to given question;

5X+8y=5   ……..eqn 1.

3x-2y=3    ………eqn 2.

Multiply by 4 in the eqn 2.

We get

12x-8y=12 ……..eqn 3 .

When we add eqn 1 and eqn 3 we get ;

17x =17

X =1

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After eliminating the y-term in the given equations i.e 5x+8y=5 and

3x-2y=3, The solution to the system of equations is x = 1, y = 0.

The elimination method is the technique used to solve systems of linear equations. It involves adding or subtracting equations in a system to eliminate one variable and find the value of the other variable.

To eliminate the y-term in these equations, you can multiply the second equation by 4, which will give you:

12x - 8y = 12

Now, you can add this equation to the first equation:

5x + 12x + 0y = 5 + 12

Simplifying the left side gives you 17x, and simplifying the right side gives you 17. So, you have the equation:

17x = 17

Solving for x gives you x = 1.

Now that you have found the value of x, you can substitute it into one of the original equations to solve for y. Using the first equation, you get:

5(1) + 8y = 5

Simplifying this equation gives you:

8y = 0

Solving for y gives you y = 0.

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Random sampling is crucial when surveying as it ensures that the sample selected is representative of the population.

By randomly selecting participants from the population, the sample is likely to be unbiased on average, which means that the results of the study can be generalized to the entire population. Without random sampling, the results of the study may be skewed or biased towards a certain group, which can lead to incorrect conclusions and poor decision-making. Therefore, it is essential to use random sampling when surveying to obtain accurate and reliable results.

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With four decimal places added, we have P(2.04 Z 3.04) 0.0189.

To round a decimal value to two decimal places, use the hundredths place, which is the second place to the right of the decimal point.

Subtracting the area to the left of 1.25 from the area to the left of 2.15 will give us the standard normal area between 1.25 and 2.15.

The area to the left of 1.25 is 0.8944, and the area to the left of 2.15 is 0.9842, according to a conventional normal distribution table or calculator.

So, the standard normal area between 1.25 and 2.15 is:

P(1.25 < Z < 2.15) = 0.9842 - 0.8944 = 0.0898

Rounding to four decimal places, we get:

P(1.25 < Z < 2.15) ≈ 0.0898

We follow the same procedure as before to determine the standard normal region between 2.04 and 3.04:

P(2.04 < Z < 3.04) = P(Z < 3.04) - P(Z < 2.04)\

The area to the left of 2.04 is 0.9798, and the area to the left of 3.04 is 0.9987, according to a conventional normal distribution table or calculator.

So, the standard normal area between 2.04 and 3.04 is:

P(2.04 < Z < 3.04) = 0.9987 - 0.9798 = 0.0189

Rounding to four decimal places, we get:

P(2.04 < Z < 3.04) ≈ 0.0189

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The approximate volume of a human head is 4000 cubic centimeter.

A sphere is simply a geometrical object that is a three-dimensional analogue to a two-dimensional circle.

The circumference of a sphere is given by the formula:

C = 2πr

Where r is the radius of the sphere. In this case, we are given that the circumference of the sphere (which approximates the human head) is 60 cm, so:

60 cm = 2πr

Solve for r

r = 60/2π

r = 30/π

The volume of a sphere is given by the formula:

V = (4/3)πr³.

Substituting the value we found for r, we get:

V = (4/3) × π × ( 30/π )³

V = (4/3) × π × ( 30/π )³

V = 3647.566 cm³

V = 4000 cm³

Therefore, the volume to the nearest 1,000 cubic cm is 4000 cm³.

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Step-by-step explanation:

To find the range, we need to subtract the smallest value from the largest value in the dataset:

Range = Largest value - Smallest value

Range = 92 - 3

Range = 89

To find the variance and standard deviation, we need to calculate the mean first:

Mean = (Sum of all values) / (Number of values)

Mean = (92+19+41+24+75+53+70+3+67+64+9) / 11

Mean = 45.09 (rounded to two decimal places)

Next, we need to calculate the variance:

Variance = (Sum of squared differences from the mean) / (Number of values - 1)

Variance = [(92-45.09)^2 + (19-45.09)^2 + (41-45.09)^2 + (24-45.09)^2 + (75-45.09)^2 + (53-45.09)^2 + (70-45.09)^2 + (3-45.09)^2 + (67-45.09)^2 + (64-45.09)^2 + (9-45.09)^2] / (11-1)

Variance = 1071.45 (rounded to two decimal places)

Finally, we can calculate the standard deviation by taking the square root of the variance:

Standard deviation = Square root of variance

Standard deviation = Square root of 1071.45

Standard deviation = 32.74 (rounded to two decimal places)

The range tells us the difference between the highest and lowest values in the dataset, which in this case is 89. The variance and standard deviation tell us how spread out the data is from the mean. The higher the variance and standard deviation, the more spread out the data is. In this case, the variance and standard deviation are both relatively high, indicating that the data is fairly spread out.

Step-by-step explanation:

So first of all u have to convert the metres into centimetres. After the conversation draw the map of both in centimetres and your map is done. The question is answered

Answer:

Determine the dimensions of the room: Measure the length and width of the room you want to draw a floor plan for using a tape measure. Record these measurements in meters.

Choose a scale: Since you want to use a scale of 1cm to 0.5m, you need to convert your measurements from meters to centimeters. For example, if your room measures 6 meters by 4 meters, you need to multiply each measurement by 100 to get 600cm by 400cm. Then, divide each measurement by 2 to get your scale measurement. In this case, your floor plan will be 300cm by 200cm.

Draw a rough sketch: Using a pencil and graph paper, draw a rough sketch of the room's shape based on the dimensions you have recorded.

Add doors and windows: Using the same scale, add doors and windows to your floor plan. Doors are typically represented by a straight line with an arc on top, while windows are represented by a straight line with a horizontal line through the middle.

Add fixtures and appliances: Add any fixtures and appliances that are permanent to the room, such as sinks, cabinets, and appliances. You can use symbols to represent these items, such as a rectangle for a refrigerator or a triangle for a sink.

Label everything: Finally, label everything on your floor plan using a legible font. This includes the dimensions of the room, the location of doors and windows, and the names of fixtures and appliances.

Step-by-step explanation:

Since means cannot be smaller than 0, the sampling distribution of the mean is always right skewed.

No matter the sample size, the form of the sampling distribution of means is always the same as the population distribution.

We require two assumptions in order to apply the sampling distribution model to sample proportions: The selected values must be independent of one another, according to the independence assumption. The Sample Size Assumption demands that the sample size, n, be sufficiently large.

While doing a t-test, it is typical to make the following assumptions: the measuring scale, random sampling, normality of the data distribution, sufficiency of the sample size, and equality of variance in standard deviation.

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the actual question is :

Which of the following is true about the sampling distribution of the mean?

a. It is an observed distribution of scores

b. It is a hypothetical distribution

c. It will tend to be normally distributed with a

standard deviation equal to the population

standard deviation

d. The mean will be estimated by the standard

error

e. Both (a) and (b)

Answer:

FIRST ONE "Deb sold vases for two years, neither sold nor bought the next year and then sold bases for two more years"

Step-by-step explanation:

Notice the number of bases in debs collection is DECREASING as the years passes for the first and third period. This is she is selling her vases but in the middle the number is the same (two point in the same horizontal line) this means she neither sold nor bought any vase in that period.

Answer:

Step-by-step explanation:

The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle. Solving for the radius, we have:

r = C/(2π) = 50.24/(2π) ≈ 8

So the radius of the circle is approximately 8 units.

The formula for the area of a circle is A = πr^2, so we have:

A = π(8)^2 = 64π ≈ 201.06

Therefore, the area of the circle is approximately 201.06 square units.

A defective stamp is likely to be chosen at random 1.4% of the time, or about 0.014 times. The observed percentage probability  faulty stamps is below the 3.5% maximum permitted rate,

Name an event from one outcome. Step 2: Compile a list of all potential outcomes, including any positive ones. Step 3: Subtract the number of favorable outcomes from the total number of possibilities that are feasible.

P(faulty) = quantity of defective stamps divided by total quantity of stamps

P(defective) = 10/700.

0.014 P(defective)

Consequently, there is a 1.4% chance (or about 0.014) that a randomly chosen stamp will be flawed.

B-The observed percentage of flawed stamps is:

10 / 700 0.5 – 0.014

Divide this rate by 100 to get the percentage:

0.014 x 100 approximately 1.4%

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The ratio between the number of red gummy bears to the number of yellow gummy bears is of:

20/23.

The ratio between the number of red gummy bears and the number of yellow gummy bears is obtained applying the proportions in the context of the problem.

To obtain the ratio between two amounts A and B, you need to divide the first amount by the second amount. The result of this division will give you the ratio of the two amounts.

The amounts for this problem are given as follows:

Hence the ratio between these two amounts is given as follows:

20/23.

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The equation of the line is y = -2x. The correct option is the last option y = -2x

From the question, we are to write the equation of the line in the given graph using the slope and y-intercept from the graph

First, we will determine the slope of the graph

The slope of the graph calculated from the formula

Slope = Change in y / Change in x

Slope = (y₂ - y₁) / (x₂ - x₁)

Picking the points (-1, 2) and (0, 0)

Slope = (0 - 2) / (0 - (-1))

Slope = -2/(0 + 1)

Slope = -2/1

Thus,

Slope = -2

From the graph, the y-intercept of the graph is 0

Then,

From the slope-intercept form of the equation of a line,

y = mx + c

Where m is the slope

and c is the y-intercepts

The equation of the line is

y = -2x + 0

y = -2x

Hence, the equation is y = -2x

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Answer:

m = 0

Step-by-step explanation:

Slope = rise/run or (y2 - y1) / (x2 - x1)

Pick 2 points (0,2) (1,2)

We see the y stay the same and the x increase by 1, so the slope is

m = 0/1 = 0

So, the slope is 0

The total amount of money Carol originally invested is £22,000 in the bank account.

Compound interest is interest that is calculated not only on the initial amount of money invested or borrowed, but also on any accumulated interest from previous periods.

This results in exponential growth or accumulation of interest over time.

Let X be the amount that Carol originally invested in the account.

After 1 year, the amount of money in the account will be X(1+0.025) = X(1.025).

After Carol withdrew £1000, the amount of money in the account will be X(1.025) - £1000.

After 2 years (i.e. on 1st January 2016), the amount of money in the account will be (X(1.025) - £1000)(1+0.025) = (X(1.025) - £1000)(1.025).

We know that the amount of money in the account on 1st January 2016 was £23,517.60, so we can write the equation:

(X(1.025) - £1000)(1.025) = £23,517.60

Expanding the left-hand side and simplifying, we get:

X(1.025)² - £1000(1.025) = £23,517.60

X(1.025)² = £24,567.63

Dividing both sides by (1.025)², we get:

X = £22,000 (rounded to the nearest pound)

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The complete question is -

On the 1st of January 2014, Carol invested some money in a bank account. The account pays 2.5% compound interest per year. On the 1st of January 2015, Carol withdrew £1000 from the account. On the 1st of January 2016, she had £23 517.60 in the account. Work out how much Carol originally invested in the account?

Answer:

150

Step-by-step explanation:

Using the formulas

A=6a2

V=a3

Solving forA

A=6V⅔=6·125⅔ ≈150

Answer:

Step-by-step explanation:

If the volume of a cube is 125 cm³, it means that each side of the cube measures 5 cm (since 5 x 5 x 5 = 125).

To find the surface area of the cube, we need to calculate the area of each of the six faces and add them together.

The area of each face is simply the length of one side squared (or side x side).

So, the surface area of the cube would be:

6 x (5 cm x 5 cm) = 6 x 25 = 150 cm²

Therefore, the surface area of the cube is 150 cm².

The perimeter of the given figures is: Triangle = 4x - 2. Rectangle = 8x - 8, and square = 12x - 8y.

The whole distance encircling a form is referred to as its perimeter. It is the length of any two-dimensional geometric shape's border or outline. Depending on the size, the perimeter of several figures can be the same. Consider a triangle built of an L-length wire, for instance. If all the sides are the same length, the same wire can be used to create a square.

The perimeter of a figure is the sum of all the segments of the figure.

The perimeter of triangle is:

P = 2x - 5 + x + x + 3 = 4x - 2

The perimeter of rectangle is:

P = 2(l + b)

P = 2(3x + 1 + x - 5)

P = 2(4x - 4)

P = 8x - 8

The perimeter of square is:

P = 4(s)

P = 4(3x - 2y)

P= 12x - 8y

Hence, the perimeter of the given figures is: Triangle = 4x - 2. Rectangle = 8x - 8, and square = 12x - 8y.

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The probability of obtaining a reading between -0.08°C and 1.68°C is approximately 0.4854 or 48.54%.

Probability is the branch of mathematics concerned with the occurrence of a random event, and there are four types of probability: classical, empirical, subjective, and axiomatic.

The readings at freezing on a set of thermometers are normally distributed, with a mean () of 0°C and a standard deviation () of 1.00°C. We want to know how likely it is that we will get a reading between -0.08°C and 1.68°C.

To solve this problem, we must use the z-score formula to standardise the values:

z = (x - μ) / σ

where x is the value for which we want to calculate the probability, is the mean, and is the standard deviation.

The lower bound is -0.08°C:

z1 = (-0.08 - 0) / 1.00 = -0.08

1.68°C is the upper bound:

z2 = (1.68 - 0) / 1.00 = 1.68

We can now use a standard normal distribution table or calculator to calculate the probabilities for each z-score.

The probability of obtaining a z-score of -0.08 or less is 0.4681, and the probability of obtaining a z-score of 1.68 or less is 0.9535, according to the table. We subtract the probability associated with the lower bound from the probability associated with the upper bound to find the probability of obtaining a reading between -0.08°C and 1.68°C:

P(-0.08°C x 1.68°C) = P(z1 z z2) = P(z 1.68) minus P(z -0.08) = 0.9535 - 0.4681 = 0.4854

As a result, the chance of getting a reading between -0.08°C and 1.68°C is approximately 0.4854 or 48.54%.

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The required value of the function  (f + g)(x) for given f(x) and g(x) as ( 3 / √x )  - ( 2 / x³ ) and √(5x - 7) is equals to ( 3 / √x )  - ( 2 / x³ )  + √(5x - 7).

Function f(x) is equals to,

( 3 / √x )  - ( 2 / x³ ) for all x > 0

Function g(x) is equals to,

g(x) = √(5x - 7)

To get the value of (f + g)(x),

Substitute the value of f(x) and g(x) and  add the functions f(x) and g(x) together,

Sum of f(x) and g(x) is equals to,

(f + g)(x)

= f(x) + g(x)

= ( 3 / √x )  - ( 2 / x³ )  + √(5x - 7)

Therefore, value of the function (f + g)(x) is equals to ( 3 / √x )  - ( 2 / x³ )  + √(5x - 7).

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The above question is incomplete, the complete question is:

The function f is defined by f of x is equal to 3 divided by the square root of x minus 2 divided by x cubed for x > 0, g as a function of x is equal to the square root of quantity 5 x minus 7 Find (f + g)(x).

Answer:

To solve the quadratic equation 9×^2-15×-6=0, we can use the quadratic formula, which is given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

In this case, a = 9, b = -15, and c = -6, so we can substitute these values into the quadratic formula:

x = (-(-15) ± sqrt((-15)^2 - 4(9)(-6))) / 2(9)

Simplifying this expression gives:

x = (15 ± sqrt(225 + 216)) / 18

x = (15 ± sqrt(441)) / 18

x = (15 ± 21) / 18

So the two solutions to the quadratic equation are:

x = (15 + 21) / 18 = 2

x = (15 - 21) / 18 = -1/3

Therefore, the solutions to the quadratic equation 9×^2-15×-6=0 are x = 2 and x = -1/3.

The probability of landing on yellow is 0.2, probability of component is 0.8, and sum of event and complement is 1.

On assuming that the pie is evenly divided into 5 parts,

So, the probability of landing on yellow is = 1/5 = 0.2,

The complement of landing on yellow is the probability of not landing on yellow, which is the probability of landing on any of the other 4 parts of the pie.

So, the probability of the complement is = 4/5 = 0.8,

The sum of the event (landing on yellow) and the complement (not landing on yellow) is equal to the probability of the entire sample space, which is 1.

⇒ P(Yellow) + P(Not Yellow) = 1

⇒ 0.2 + 0.8 = 1

So, the sum of the event and the complement is 1 or 100%.

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The given question is incomplete, the complete question is

A circular pie is divided in 5 parts , Green , Yellow, Blue Black and Red.

Find the probability of landing on yellow, the probability of complement, and the sum of the event and the complement.

The ordered pairs of given equation are (3/2,4), (1/3, -3),(5/6,0)

An ordered pair is composed of the ordinate and the abscissa of the x coordinate, with two values supplied in parentheses in a specified sequence. Placing a point on the Cartesian plane could be beneficial for visual comprehension.

for example, the ordered pair (x, y) signifies an ordered pair in which 'x' is referred to as the first element and 'y' is referred to as the second element. These items, which can be either variables , have distinct names depending on the context in which they are used. In an ordered pair, the element order is quite significant.

Given Equation of Y=6x−5

First Ordered pair;(,4)

y=4

x=4+5/6

x=3/2

First Ordered pair;(, -3)

y=-3

x=-3+5/6

x=1/3

First Ordered pair; (,0)

y=0

x=5/6

The ordered pairs of given equation are

(3/2,4), (1/3, -3),(5/6,0)

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The complete question is:

For The Equation, Y=6x−5

Complete The Given Ordered Pairs (,4), (, -3), (,0)

x = 4.9

Solution:

We can use the intersecting chords formula:

[tex]7\times7 = 10x[/tex]

[tex]49 = 10x[/tex]

[tex]49\div10=10x\div10[/tex]

[tex]4.9 = x[/tex]

Therefore, x = 4.9.

The required probabilities of catching or not catching a fish is given by,

Catching fish in all 4 lakes = 0.0048

Not catching any fish at all = 0.2688

Catching at least one fish  = 0.7312

Catching fish in Lake L1 and lake L2 = 0.12

Catching fish in lake L1 or lake L2 = 0.58

Catching fish in exactly one lake = 1.1

Probability of catching fish in all 4 lakes,

Simply multiply the probabilities of catching fish in each lake,

since the events are independent,

P(C1 and C2 and C3 and C4)

= P(C1) x P(C2) x P(C3) x P(C4)

= 0.3 x 0.4 x 0.2 x 0.2

= 0.0048

= 0.48%

Probability of not catching any fish at all,

Probability of the complement of the event of catching at least one fish.

Happy if we catch at least one fish, the probability of not catching any fish is the probability that is not happy,

P(not happy)

= P(not C1 and not C2 and not C3 and not C4)

= (1 - P(C1)) x (1 - P(C2)) x (1 - P(C3)) x (1 - P(C4))

= 0.7 x 0.6 x 0.8 x 0.8

= 0.2688

= 26.88%

Probability of catching at least one fish,

Probability of the complement of the event of not catching any fish and subtract it from 1,

P(at least one fish)

= 1 - P(not happy)

= 1 - 0.2688

= 0.7312

= 73.12%

Probability of catching fish in Lake L1 and lake L2,

Simply multiply the probabilities of catching fish in each lake,

P(C1 and C2)

= P(C1) x P(C2)

= 0.3 x 0.4

= 0.12

= 12%

Probability of catching fish in lake L1 or lake L2,

Add the probabilities of catching fish in each lake,

And then subtract the probability of catching fish in both lakes to avoid double counting,

P(C1 or C2)

= P(C1) + P(C2) - P(C1 and C2)

= 0.3 + 0.4 - 0.12

= 0.58

= 58%

Probability of catching fish in exactly one lake can be broken down into four mutually exclusive events,

Catching fish in L1 only, catching fish in L2 only, catching fish in L3 only, or catching fish in L4 only.

Probabilities of each of these events is,

P(C1 or C2 or C3 or C4)

= P(C1) +  P(C2) + P(C3) + P(C4)

=  0.3 + 0.4 + 0.2 + 0.2

= 1.1

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the Fourier series solution is not directly used to find the general solution but is used as a part of it, along with the radial solution. The Fourier series solution helps in finding the solution to the given boundary value problem, which, when combined with the radial solution, gives the complete solution to the Laplace equation.

The Fourier series approach that you have used helps in finding the solution to the boundary value problem, i.e., finding u(a,ϕ) for the given boundary conditions. However, to find a general solution to the Laplace equation, we need to use the superposition principle, which states that the sum of any two solutions to the Laplace equation is also a solution.

Therefore, we can use the previously obtained Fourier series solution for u(a,ϕ) as a building block to construct the general solution. We know that the Laplace equation has radial symmetry, which means that the temperature distribution is only a function of radius (rho) and not of angle (ϕ). Hence, we can write the general solution as:

u(rho,ϕ) = f(rho) + u(a,ϕ)

where f(rho) is the radial component of the solution and u(a,ϕ) is the previously obtained Fourier series solution.

To find f(rho), we need to solve the radial ODE using the boundary conditions at rho=0 and rho=a. Once we have obtained f(rho), we can add it to u(a,ϕ) to get the general solution.

Therefore, the Fourier series solution is not directly used to find the general solution but is used as a part of it, along with the radial solution. The Fourier series solution helps in finding the solution to the given boundary value problem, which, when combined with the radial solution, gives the complete solution to the Laplace equation.

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The three Pythagorean trigonometric identities, which I’m sure one can find in any Algebra-Trigonometry textbook, are as follows:

sin² θ + cos² θ = 1

tan² θ + 1 = sec² θ

1 + cot² θ = csc² θ

where angle θ is any angle in standard position in the xy-plane.

Consistent with the definition of an identity, the above identities are true for all values of the variable, in this case angle θ, for which the functions involved are defined.

The Pythagorean Identities are so named because they are ultimately derived from a utilization of the Pythagorean Theorem, i.e., c² = a² + b², where c is the length of the hypotenuse of a right triangle and a and b are the lengths of the other two sides.

This derivation can be easily seen when considering the special case of the unit circle (r = 1). For any angle θ in standard position in the xy-plane and whose terminal side intersects the unit circle at the point (x, y), that is a distance r = 1 from the origin, we can construct a right triangle with hypotenuse c = r, with height a = y and with base b = x so that:

c² = a² + b² becomes:

r² = y² + x² = 1²

y² + x² = 1

We also know from our study of the unit circle that x = r(cos θ) = (1)(cos θ) = cos θ and y = r(sin θ) = (1)(sin θ) = sin θ; therefore, substituting, we get:

(sin θ)² + (cos θ)² = 1

1.) sin² θ + cos² θ = 1 which is the first Pythagorean Identity.

Now, if we divide through equation 1.) by cos² θ, we get the second Pythagorean Identity as follows:

(sin² θ + cos² θ)/cos² θ = 1/cos² θ

(sin² θ/cos² θ) + (cos² θ/cos² θ) = 1/cos² θ

(sin θ/cos θ)² + 1 = (1/cos θ)²

(tan θ)² + 1 = (sec θ)²

2.) tan² θ + 1 = sec² θ

Now, if we divide through equation 1.) by sin² θ, we get the third Pythagorean Identity as follows:

(sin² θ + cos² θ)/sin² θ = 1/sin² θ

(sin² θ/sin² θ) + (cos² θ/sin² θ) = 1/sin² θ

1 + (cos θ/sin θ)² = (1/sin θ)²

1 + (cot θ)² = (csc θ)²

3.) 1 + cot² θ = csc² θ