The correct answer to the question is D. Yes, because 0.0327% is not included in the confidence interval.
How to calculate the valueConfidence interval = (sample proportion - Z * standard error of the proportion, sample proportion + Z * standard error of the proportion)
Substituting these values into the formula for the standard error of the proportion, we get:
standard error of the proportion = ✓(0.0317*(1-0.0317))/420000)
= 0.0000072
Substituting this value into the formula for the confidence interval, we get:
Confidence interval = (0.0317 - 1.96 * 0.0000072, 0.0317 + 1.96 * 0.0000072)
Therefore, the 95% confidence interval estimate of the percentage of cell phone users who develop cancer of the brain or nervous system is 0.0316% to 0.0318%.
Therefore, cell phone users appear to have a higher rate of cancer of the brain or nervous system than those who do not use cell phones. The correct answer to the question is D. Yes, because 0.0327% is not included in the confidence interval.
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what is the distance between (5, 1), (5.-6)
4.
Find the area of the shaded region.
A. 5x2 + 16x – 21
B. 7x2 + 28x + 27
C. 5x2 + 28x + 27
D. 5x2 + 28x + 21
Answer:
yay
Step-by-step explanation:
yayyayyayyayyayyyayyayayayyayaya
Answer:
D. 5x2 + 28x + 21
Step-by-step explanation:
First you want to find the area of the white rectangle
Then find the area of the beings rectangle
Then subtract beings minus white.
Sorry it’s +21. The first answer in your list
then thats correct
...............................................................................................................................................
5x^2+28x+21 confirmed from Cortez himself
consider the points p such that the distance from p to as21, 5, 3d is twice the distance from p to bs6, 2, 22d. show that the set of all such points is a sphere, and find its center and radius.
The center of the sphere can be found by taking the average of the coordinates of points A and B, and the radius of the sphere is half the distance between points A and B.
Let's denote point A as (2, 1, 5) and point B as (6, 2, 22). We want to find the set of points P such that the distance from P to A is twice the distance from P to B.
We can express this condition mathematically as:
|PA| = 2|PB|
Using the distance formula, we can rewrite this equation as:
[tex]\sqrt{(x - 2)^2+ (y - 1)^2 + (z - 5)^2} = 2\sqrt{(x - 6)^2 + (y - 2)^2 + (z - 22)^2}[/tex]
Simplifying the equation, we have:
[tex](x - 2)^2 + (y - 1)^2 + (z - 5)^2 = 4((x - 6)^2 + (y - 2)^2 + (z - 22)^2)[/tex]
Expanding and combining like terms, we get:
[tex]3x^2 + 3y^2 + 3z^2 - 28x - 10y - 96z = 273[/tex]
This equation represents the equation of a sphere. The center of the sphere can be found by taking the average of the coordinates of points A and B, which gives us the center (4, 1.5, 13.5). The radius of the sphere is half the distance between points A and B, which is the square root of 273 divided by 2.
Therefore, the set of all points satisfying the given condition is a sphere with center (4, 1.5, 13.5) and radius √(273/2).
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2
John, 25, just started contributing to his employer's 401(k) plan and started an IRA account. He is choosing to invest only in bonds and secure
Investments instead of high-risk stocks. Is John pursuing a wise investment strategy?
Select the best answer from the choices provided.
O A Yes; retirement Income should be protected from turmoil in the stock market.
OB. Yes; as john gets closer to retirement, he can switch to higher yielding stocks.
OC. No; young investors can afford more risk since they have time to recover from any sudden loses.
OD
No; bonds and safe investments are not part of good retirement plans.
Answer:
I think its C
Step-by-step explanation:
Suppose you bought a sofa for a tota purchase price of $1,254.07. State taxes were 7%. What was the amount or the sales tax?
The amount of sales tax is $87.79.
Given a total purchase price of a sofa as $1,254.07 and state taxes of 7%.
We are required to calculate the amount of sales tax.
The amount of sales tax can be calculated by multiplying the purchase price by the sales tax rate.
Let's represent the sales tax rate by `r`.
Therefore, the sales tax formula is expressed as:
Sales tax = r * purchase price
In this case, the rate of the sales tax `r` is 7%.
Therefore, we have:r = 7% = 0.07
Now we substitute the values given into the formula:
Sales tax = 0.07 * $1,254.07= $87.79
Therefore, the amount of sales tax is $87.79.
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last question 100 pts pleasee
Sum of two interiors =Exterior
Use simple way
Angle beside x
180-(30+14)180-44136Measure of x
180-13644°Answer:
x = 44°
Step-by-step explanation:
According to the Exterior Angle Theorem of a triangle, the external angle is equal to the sum of the two interior opposite angles of a triangle.
This is because the interior angles of a triangle sum to 180° and angles on a straight line sum to 180°.
⇒ x = ∠CAB + ∠BCA
⇒ x = 14° + 30°
⇒ x = 44°
Proof
Interior angles of a triangle sum to 180°
⇒ ∠CAB + ∠ABC + ∠BCA = 180°
⇒ 14° + ∠ABC + 30° = 180°
⇒ ∠ABC = 180° - 14° - 30°
⇒ ∠ABC = 136°
Angles on a straight line sum to 180°
⇒ ∠ABC + x = 180°
⇒ 136° + x = 180°
⇒ x = 180° - 136°
⇒ x = 44°
Meg used this table to track the average attendance from month to month at her theater for five months. A 5-column table with 1 row is titled Percent Increase and Decrease. Column 1 is labeled October with entry negative 1. Column 2 is labeled November with entry 0.5. Column 3 is labeled December with entry negative 0.9. Column 4 is labeled January with entries three-fourths. Column 5 is labeled February with entry one-half. A number line going from negative 1 to positive 1 in increments of 0.25. Graph the numbers on the number line. Choose the inequality symbol that makes each comparison true. Nov. Feb. Oct. Dec. Jan. Nov.
Answer:
B - Her monthly average would have increased by $19.57. ... Katie works as a waitress and records her monthly tips in the table shown below. If Katie decided not to work the month of November, how would her five month average compare to her six month ... Lisa is currently taking physics as one of her electives in school.
Step-by-step explanation:
Answer:
Nov. ✔ = Feb.
Oct. ✔ < Dec.
Jan. ✔ > Nov.
Step-by-step explanation:
Edge 2022
How does an online discussion group differ from one that is held face-to-face?
There is no need for group roles when holding online meetings.
Online discussion groups do not allow members to develop critical thinking skills.
Group members are more likely to bond together as a group online than they would when meeting face-to-face.
Facilitating an online conversation may require even more work, so that participants continue engaging with one
another online.
Answer: Online discussions have a better portrayal of other cultures
Step-by-step explanation:
Unless you are very antisocial then the only good thing that an online group shares is being able to stay safe from corona virus and the ability to speak with each other from different perspectives. Lets say hypothetically I was in a bear watching group where everybody took pictures and discussed bears. If I lived in North Michigan I would see black bears and brown bears. If I lived in more of an arctic region I would see polar bears. You can see how this would throw in a little more content to discuss. Which bears are bigger? Which bears live longer? Which bears have heavier coats? This obviously does not only apply to bears. Books, sports, video games, cars, and more all have different miniature cultures in different states or countries.
Answer:
Facilitating an online conversation may require even more work, so that participants continue engaging with one another online.
Step-by-step explanation:
A family of 10 purchased tickets to the county fair. Tickets for adults cost $6 and tickets for children cost $3. If the total cost of the tickets was $42, how many family members were adults and children?
The family members were 3 adults and 2 children.
We have given that the,
A family of 10 purchased tickets to the county fair.
Tickets for adults cost $6 and tickets for children cost $3
Total cost is $42.
We have to determine how many family members were adults and children
What is the next step from the given condition?
(3 x 10) + (2 x 6)
30+12=42
Therefore we get 3 adults and 2 children.
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Assume that C(x) is in dollars and x is the number of units produced and sold. For the total cost function C(x)=0.01x + 0.4x + 50, find AC and C'(x) when x = 90 and Ax=1 AC =$ 1 (Simplify your answer. Type an integer or decimal rounded to two decimal places as needed.) c'(x) when x = 90 is $ . (Type an integer or a decimal.)
The average cost AC when x = 90 is $1, and the derivative C'(x) when x = 90 is $0.41.
The average cost function AC and the derivative of the total cost function C'(x) can be found using the given total cost function C(x) = 0.01x + 0.4x + 50, where x represents the number of units produced and sold.
To find the average cost AC, we divide the total cost C(x) by the quantity x:
AC = C(x) / x
Substituting the given total cost function C(x) = 0.01x + 0.4x + 50, we have:
AC = (0.01x + 0.4x + 50) / x
Simplifying, we get:
AC = (0.41x + 50) / x
When x = 90, we substitute this value into the equation:
AC = (0.41 * 90 + 50) / 90
AC = (36.9 + 50) / 90
AC = 86.9 / 90
AC ≈ $0.97 ≈ $1 (rounded to two decimal places)
To find the derivative C'(x), we differentiate the total cost function C(x) with respect to x:
C'(x) = d/dx (0.01x + 0.4x + 50)
C'(x) = 0.01 + 0.4
C'(x) = 0.41
When x = 90, we substitute this value into the equation:
C'(90) = 0.41
Therefore, the derivative C'(x) when x = 90 is $0.41.
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Which of the following has the greatest value.
Answer:
A.
Step-by-step explanation:
The first question equals 729 and it is the highest, I reccomend using photomath.
A statue casts a shadow that is 9 ft long. A boy that is 4 ft tall casts a shadow that is 6 ft long. How tall is the statue?
Answer:
6 feet tall.
Step-by-step explanation:
The boy is 4 ft tall, but he casts a 6 ft long shadow. You need to find how long a shadow would be if a person/thing was just 1 ft tall.
To do this, you would need to divide 4 by 4, and 6 by 4 as well.
6/4 is 1.5, or 1 and a half.
If a person (or a thing) is 1 foot tall, the statue will cast 1 and a half feet long shadow.
The statue casts a 9 ft long shadow.
Since we already figured out 1 ft = 1 and 1/2 ft shadow, and since the statue casts a 9 foot long shadow, you need to divide 9 by 1.5.
(You'll get 6.)
This means that the statue is 6 feet tall, and it casts a 9 foot long shadow.
please help me it would help a lot please <33
Answer:
11
Step-by-step explanation:
first u 90 minus 67 equals 23
and if u times 11 by 4 equal 44
minus 21 equals 23 which would make the equation true
hope this helped
The mean is _____ than the median, which in turn is _____ the mode, in a unimodal skewed right distribution.
a. less, greater than or equal to
b. less, less than
c. greater, less than
d. greater, greater than or equal to
The mean is greater than the median, which in turn is less than the mode, in a unimodal skewed right distribution. Therefore, option c) is the correct answer.
The terms mean, median, and mode are commonly used in statistics to measure the central tendency of a set of
values or a dataset. The mean is calculated by dividing the sum of all the numbers in a dataset by the total number of
items in the dataset. The mean is the average of the dataset. The median is the middle number in a dataset when the
data is arranged in ascending or descending order. Half of the values are higher than the median, and half are lower.
The mode is the value that appears most frequently in a dataset. If there are two values that occur with the same
frequency, the dataset is referred to as bimodal, and if there are more than two values that occur with the same
frequency, the dataset is referred to as multimodal. In a unimodal skewed right distribution, the mean is greater than
the median, which in turn is less than the mode. Therefore, the correct answer is option c) greater, less than.
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Decide whether it is possible for a triangle to have the three angle measures or three side lengths given.
If it is possible, then decide whether all such triangles are congruent.
Answer:
Step-by-step explanation:
the first one cant be a triangle
im pretty sure the second one can
the third one can
im pretty sure the fourth one cant
Can I get help with number 14
Answer:
A
Step-by-step explanation:
-y = -2x+5
y = 2x-5
3x+2(2x-5)= -3
Together donkey and Shrek can make 7 cupcakes in 12 mins. Shrek can make 2 cupcakes in 8 mins. How long will it take Donkey to make 3 cupcakes
Anne picked out 3 board games from her cabinet to take to a game night with her friends. She still has more than 10 board games in the cabinet, so she thinks she could bring a few more. Let x represent how many board games were in Anne’s cabinet to start. Which inequality describes the problem?
Answer:
x - 3 > 10
There were more than
13
board games in Anne's cabinet to start.
Step-by-step explanation:
The variable x represents how many board games were in Anne's cabinet to start. Since she has already picked out 3 board games to bring, the expression x–3 represents how many board games are still in the cabinet.
And, since Anne has more than 10 board games still in her cabinet, x–3 must be greater than 10.
This inequality shows the relationship.
x–3>10
Now, solve for x.
x–3
> 10
x–3+3
> 10+3 Add 3 to both sides
x
> 13 Simplify
So, there were more than 13 board games in Anne's cabinet to start.
Recall that, fixed a set U (which we call the universe of discourse), we have certain operations on subsets of U so that, for all A, B, C CU the following equivalences and equalities hold. >> AC BUC, ABCC CCA⇒B >> CnACB, A" = A, (AUB)* = A*n B', (An B)* = A*UB*. ACB →B'CA, You can answer just one of the following parts, not both. You can support your answer with drawings of Venn diagrams, but you need to give an argument according to the specifications for full credit. (a) Prove that for any given sets A, BCU, we have that B\A= (AB)* using only the above equations and equivalences. (Hint: Notice that two sets X, Y CU are equal if and only if, for every CCU, we have XCC YCC.) (b) Prove that for any given sets A, BCU, we have that B* A* = (AB)* using the definitions of the operations (). \, and in terms of the elements of U, A, and B.
(a) To prove that B\A = (AB)* using only the above equations and equivalences, we need to show that B\A is equivalent to (AB)*.
First, we will show that B\A is a subset of (AB)*.
Let x be an element of B\A. Then, x is in B and x is not in A. Therefore, x is in AB and not in A. This means that x is in (AB)*. Thus, we have shown that B\A is a subset of (AB)*.
Next, we will show that (AB)* is a subset of B\A.
Let x be an element of (AB)*. Then, x is in AB or x is not in AB.
If x is in AB, then x is in B and x is in A. Therefore, x is not in B\A.
If x is not in AB, then x is not in A or x is not in B. Therefore, x is not in A and x is in B. This means that x is in B\A.
Thus, we have shown that (AB)* is a subset of B\A.
Since we have shown that B\A is a subset of (AB)* and (AB)* is a subset of B\A, we can conclude that B\A = (AB)*.
(b) To prove that B* A* = (AB)*, we need to show that B* A* is a subset of (AB)* and (AB)* is a subset of B* A*.
First, we will show that B* A* is a subset of (AB)*.
Let x be an element of B* A*. Then, x is in (B*) and x is in (A*).
If x is in B*, then x is in B or x is not in B.
If x is in A*, then x is in A or x is not in A.
If x is in B and x is in A, then x is in AB.
If x is not in B and x is not in A, then x is not in AB.
If x is in B and x is not in A, then x is in B\A.
If x is not in B and x is in A, then x is in A\B.
Therefore, we have shown that x is in (AB)*.
Next
Find the eigenvalues and the corresponding eigenspaces for the matrix -2 0 1 1 0 -1 0 1 - 1 Here the characteristic polynomial should be cubic. You may use a calculator or Wolfram Alpha to factor the characteristic polynomial if you wish.
The eigenvalues of the matrix -2 0 1 1 0 -1 0 1 -1 are λ₁ = -1, λ₂ = 1, and λ₃ = -1. The corresponding eigenspaces are E₁ = span{[-1, 1, 0]}, E₂ = span{[1, 1, 1]}, and E₃ = span{[-1, 1, 2]}.
Eigenvalues and eigenvectors play a fundamental role in linear algebra, particularly in the study of matrices.
The eigenvalues of a matrix are the values λ for which the equation A = λ has nontrivial solutions, where A is the given matrix and is a non-zero vector. The eigenspace associated with an eigenvalue is the set of all eigenvectors corresponding to that eigenvalue.
To find the eigenvalues of the matrix -2 0 1 1 0 -1 0 1 -1, we need to solve the characteristic equation det(A - λI) = 0, where A is the given matrix, λ is an eigenvalue, and I is the identity matrix.
The characteristic equation in this case is (-2 - λ)(λ² + 1) + (1 - λ)(-1) = 0. Simplifying this equation yields λ³ - 2λ² - 2 = 0. This is a cubic equation, and we can use a calculator or Wolfram Alpha to find its roots, which are λ₁ = -1, λ₂ = 1, and λ₃ = -1.
Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation (A - λ) = 0 for each eigenvalue.
For λ₁ = -1, solving (A + ) = 0 gives us the eigenvector [-1, 1, 0]. For λ₂ = 1, solving (A - ) = 0 gives us the eigenvector [1, 1, 1].
Finally, for λ₃ = -1, solving (A + ) = 0 gives us the eigenvector [-1, 1, 2]. These eigenvectors span the eigenspaces E₁, E₂, and E₃, respectively.
In summary, the eigenvalues of the matrix -2 0 1 1 0 -1 0 1 -1 are λ₁ = -1, λ₂ = 1, and λ₃ = -1. The corresponding eigenspaces are E₁ = span{[-1, 1, 0]}, E₂ = span{[1, 1, 1]}, and E₃ = span{[-1, 1, 2]}.
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Based on the graph below, what is the domain and range of the function?
Answer:
The f(x) domain is the set of all values for the function and the function spectrum is the set of all values for which f is defined.
Step-by-step explanation:
Answer:
Domain = all real numbers
Range = y ≥ -4
Step-by-step explanation:
Domain is the x-values, and for parabolas it is almost always all real numbers.
Range is the y-values, and the lowest it gets is -4.
I NEED HELP REALLY FAST AND EXPLAIN HOW YOU SOLVED IT PLEASE
Answer:
C
Step-by-step explanation:
All you do is take the number choices and plug it into the equation. For example if you take A and plug it in the equation would look like -5(-5) < 25 and -5 times -5 is 25 so you know 25 is equal to 25 not less than.
If you plug in C you get -5(-4) < 25 and -5 times -4 is 20 and 20 is indeed less than 25
What is the radius of a circle with a diameter of 240 mm?
A) 76.43 mm
B) 480.00 mm
C)12.00 mm
D) 120.00 mm
The radius of a circle is half of its diameter so the radius of a circle with a diameter of 240 mm is 120.00 mm. Option D is the correct answer.
To find the radius of a circle with a given diameter, you can follow these steps:
Given that the diameter is 240 mm, divide it by 2 to obtain the radius. Recall that the radius is half the length of the diameter.
Radius = Diameter / 2
In this case, Radius = 240 mm / 2 = 120 mm.
Therefore, the radius of the circle with a diameter of 240 mm is 120.00 mm.
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What does this mean and how do I do it
Answer:
Step-by-step explanation:
The degree of f(x) is 0.
Its leading coefficient is 13 and the type is constant. Because the function is constant,
f(x = 13 when x --> -∞ and
f(x) = 13 when x --> ∞ .
(How? Because I'm smart like that!! :D)
- Which is the inverse or the funcion f(x)=x²-16?
Given:
The function is:
[tex]f(x)=x^2-16[/tex]
To find:
The inverse of the function.
Solution:
We have,
[tex]f(x)=x^2-16[/tex]
Step 1: Substitute [tex]f(x)=y[/tex].
[tex]y=x^2-16[/tex]
Step 2: Interchange x and y.
[tex]x=y^2-16[/tex]
Step 3: Isolate y.
[tex]x+16=y^2[/tex]
[tex]\pm \sqrt{x+16}=y[/tex]
[tex]y=\pm \sqrt{x+16}[/tex]
Step 4: Substitute [tex]y=f^{-1}(x)[/tex].
[tex]f^{-1}(x)=\pm \sqrt{x+16}[/tex]
Therefore, the inverse function of the given function is [tex]f^{-1}(x)=\pm \sqrt{x+16}[/tex].
A right rectangular prism has a length of 12 ft, width of 5 1/2 ft, and height of 10 1/2 ft. What is the volume of the prism? Enter the answer in the box.
Answer: The answer is: 693 ft^3 or 693ft cubed
Let U CC be a region containing D(0; 1) and let f be a meromorphic function on U, which has no zeros and no poles on aD(0; 1). If f has a zero at 0 and if Ref(z) > 0 for every ze aD(0; 1), show that f has a pole in D(0; 1).
It is proved that if f has a zero at 0 and if Ref(z) > 0 for every ze aD(0; 1), then f has a pole in D(0; 1).
The problem is based on Complex Analysis.
Let U CC be a region containing D(0; 1) and let f be a meromorphic function on U, which has no zeros and no poles on aD(0; 1).
If f has a zero at 0 and if Ref(z) > 0 for every ze aD(0; 1), we need to prove that f has a pole in D(0; 1).
We know that f is meromorphic function on U and we are given that it has a zero at 0. So, we can write the Laurent expansion of f around 0 as: f(z) = anzn+ ... + a1z + a0 + b1z + ... where an ≠ 0 and the expansion is valid in a deleted neighborhood of z=0. This implies that f(z) has a pole of order m at 0 where m is the largest non-negative integer such that am ≠ 0.
Suppose that m = 0 and f(z) has a removable singularity at 0 then by Riemann’s theorem f(z) is bounded in some deleted neighborhood of z = 0 which implies that Ref(z) ≤ M, a contradiction to Ref(z) > 0 for every ze aD(0; 1).
Hence, f(z) has a pole at 0. Therefore, f has a pole in D(0; 1). Hence, we have proved that if f has a zero at 0 and if Ref(z) > 0 for every ze aD(0; 1), then f has a pole in D(0; 1).
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You believe the population is normally distributed and you know the standard deviation is σ = 5.2. You obtain a sample mean of M = 78.5 for a sample of size n = 64.
What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic= p-value=
What is the p-value for this sample? (Report answer accurate to four decimal places.)
The z-score table shows that for a z-score of 15.1, the p-value is approximately zero (p < 0.0001). Hence, the p-value for this sample is p < 0.0001.
Given that the population is normally distributed and the standard deviation is σ = 5.2.
A sample of size n = 64 is obtained with the sample mean of M = 78.5.
Test statistic = (Sample mean - population mean) / (Standard error of the mean) = (78.5 - µ) / (σ /√n)
Where µ = population mean = 0σ = 5.2n = 64.
The formula for the standard error of the mean is; σM = σ/√n = 5.2/√64 = 0.65.
Substituting in the test statistic equation,
Test statistic = (78.5 - 0) / 0.65 = 121.54.
P-value is the probability of obtaining the observed sample mean or a more extreme value from the null hypothesis.
Assuming a significance level of α = 0.05 and the null hypothesis H0: µ = 0 (Population mean), we can obtain the p-value from the z-score table.z-score = (sample mean - population mean) / standard deviation = (78.5 - 0) / 5.2 = 15.1
The z-score table shows that for a z-score of 15.1, the p-value is approximately zero (p < 0.0001).Hence, the p-value for this sample is p < 0.0001.
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Which data set is the most clustered around its mean?
A.4, 10, 8, 6
B.11, 3, 10, 4
C.2, 9 ,13, 4
D.11, 2, 9, 6
The data set is the most clustered around its mean is
A.4, 10, 8, 6
How to find the dataTo determine which data set is the most clustered around its mean, we can calculate the mean and the standard deviation for each data set.
calculate the mean and standard deviation for each data set
Data Set A: 4, 10, 8, 6
Mean: (4 + 10 + 8 + 6) / 4 = 28 / 4 = 7
Standard Deviation: √[(4-7)^2 + (10-7)^2 + (8-7)^2 + (6-7)^2] / 4 ≈ 1.58
Data Set B: 11, 3, 10, 4
Mean: (11 + 3 + 10 + 4) / 4 = 28 / 4 = 7
Standard Deviation: √[(11-7)^2 + (3-7)^2 + (10-7)^2 + (4-7)^2] / 4 ≈ 3.87
Data Set C: 2, 9, 13, 4
Mean: (2 + 9 + 13 + 4) / 4 = 28 / 4 = 7
Standard Deviation: √[(2-7)^2 + (9-7)^2 + (13-7)^2 + (4-7)^2] / 4 ≈ 4.27
Data Set D: 11, 2, 9, 6
Mean: (11 + 2 + 9 + 6) / 4 = 28 / 4 = 7
Standard Deviation: √[(11-7)^2 + (2-7)^2 + (9-7)^2 + (6-7)^2] / 4 ≈ 3.27
Comparing the standard deviations, we find that data set A has the smallest standard deviation of approximately 1.58. this indicates that the data points in Data Set A are the closest to the mean, making it the most clustered around its mean.
Therefore, the answer is A. 4, 10, 8, 6.
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Use complete sentences to describe the domain of the consine function.
Answer:
The domain of a function is the set of the possible input values of the function. For example: consider the function f(x) = cos x, the domain of the function is the set of possible values of x.
The cosine function takes x values from all real numbers.
Therefore, the domain of the cosine function is a real numbers.
Step-by-step explanation: