The expression cos⁴ θ in terms of the first power of cosine is [ 3 + 2cos 2θ + cos 4θ]/8.
The power-reducing formula, for cosine, is,
cos² θ = (1/2)[1 + cos 2θ].
In the question, we are asked to use the formulas for lowering powers to rewrite the expression in terms of the first power of cosine cos⁴ θ.
We can do it as follows:
cos⁴ θ
= (cos² θ)²
= {(1/2)[1 + cos 2θ]}²
= (1/4)[1 + cos 2θ]²
= (1/4)(1 + 2cos 2θ + cos² 2θ] {Using (a + b)² = a² + 2ab + b²}
= 1/4 + (1/2)cos 2θ + (1/4)(cos ² 2θ)
= 1/4 + (1/2)cos 2θ + (1/4)(1/2)[1 + cos 4θ]
= 1/4 + cos 2θ/4 + 1/8 + cos 4θ/8
= 3/8 + cos 2θ/4 + cos 4θ/8
= [ 3 + 2cos 2θ + cos 4θ]/8.
Thus, the expression cos⁴ θ in terms of the first power of cosine is [ 3 + 2cos 2θ + cos 4θ]/8.
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I really need help with this question. I have attached two screenshots, the first is the question and the second one is the options that I have as the answer. I would really appreciate an answer!!!
Using proportions, it is found that the coordinates of point B are given as follows:
A. (1.6, 0).
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
Point B is 3/4 of the way from AB to AC, hence:
[tex]B - A = \frac{3}{4}(C - A)[/tex]
The points are:
A(-2,-3).B(x,y).C(3,1).
Then, to find the x-coordinate:
[tex]x + 2 = \frac{3}{4}(3 + 2)[/tex]
x + 2 = 3.6.
x = 1.6.
For the y-coordinate:
[tex]y + 3 = \frac{3}{4}(1 + 3)[/tex]
y + 3 = 3
y = 0.
Hence option A is correct.
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The domain of this function is {-12, -6, 3, 15}. y=-2/3x+7 complete the table based on the given domain.
If the domain of the function y=-2/3 x+7 is {-12,-6,3,15} then the range of the function is {15,11,5,-3}.
Given that the domain of the sunction y=-2/3 x+7 is {-12,-6,3,15}.
We have to find the range of the function and complete the table.
Function is relationship between two or more variables expressed in equalto form. In a function each value of x has some value of y.
To complete the table we need to find the value of the function at different values of x.Domain is basically value of x which satisfies the function.
We have to first put the value of x=-12 in y=-2/3+7
y--2/3*-12+7
=15
Now put x=-6 in the function.
y=-2/3 *-6+7
=3
Put the value of x=3 in the function.
y=-2/3*3+7
=5
Put the value of x=15.
y=-2/3*15+7
=-41/3.
Hence the range of function is {15,11,-41/2,5}.
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Sheila has a plan to save $45 a month for 18 months so that she has $810 to remodel her bathroom. After 13 months Sheila has saved $510. If the most Sheila can possibly save is $70 per month, which of the following statements is true?
a.
Sheila will meet her goal and does not need to adjust her plan.
b.
Sheila must save $50 per month to achieve her goal.
c.
Sheila must save $60 per month to achieve her goal.
d.
Sheila will not be able to achieve her goal.
The statement that is true about Sheila's savings plan is that; C.
Sheila must save $60 per month to achieve her goal.
What is true about the savings plan?
The price of bathroom remodeling is $ 810.
After 13 months , she has saved $ 510
This means that she has a $300 shortage to cover in 5 months.
This means that she need to save :
$300/5 = $60 / month in order to achieve her goal
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Answer: C
Step-by-step explanation: edge
The vertices of a quadrilateral are A(-1,6),B(-2,4),C(2,2), and D(3,4). Write a paragraph proof to determine whether quadrilateral ABCD is a rectangle. 15px
Determining the slopes of each side, we get the slope of [tex]\overline{AB}[/tex] is [tex]2[/tex], the slope of [tex]\overline{BC}[/tex] is -1/2, the slope of [tex]\overline{CD}[/tex] is 2, and the slope of [tex]\overline{AD}[/tex] is -1/2. Since the slopes of sides AB and CD and BC and AD are equal, it follows that [tex]\overline{AB} \parallel \overline{CD}[/tex] and [tex]\overline{BC} \parallel \overline{AD}[/tex]. Thus, ABCD is a parallelogram because it is a quadrilateral with two pairs of opposite congruent sides. However, we can also note that the slope of side AB is the negative reciprocal of that of sides BC, and thus [tex]\overline{AB} \perp \overline{BC}[/tex]. Using the fact that perpendicular lines form right angles, we can conclude that [tex]\angle ABC[/tex] is a right angle, and since ABCD is thus a parallelogram with a right angle, it must also be a rectangle.
Answer:
Answers will vary based on the method used, but the final slope-intercept form of the equation of AB↔ will be the same.Let the equation of AB↔ in slope-intercept form be y=mx+d, where m is the slope and d is the y-intercept.The coordinates of points A and B are (1, 1) and (5, 3), respectively, so the slope of AB↔ is m=yB−yAxB−xA=24=0.5.Substitute the value of m back in the equation:y = 0.5x + d.Substitute the coordinate of point A (1, 1) in the equation above, and solve for d: 1 = 0.5 + dd = 0.5.Therefore, the equation of AB↔ is y = 0.5x + 0.5. To check whether C lies on AB↔ substitute the x-coordinate of C into the right side of the equation: 0.5 (2.6) + 0.5 = 1.8.The result is equal to the y-coordinate of C. Thus, C satisfies the equation of AB↔ which means that C lies on AB↔.
Step-by-step explanation:
I did it
Xy has 10 pets: 4 dogs, 3 cats, 2 alpacas and 1 bunny. If he wants to arrange them in a row and make sure the pets are always grouped according to species, how many ways can he arrange the pets?
The number of ways pets can be arranged in group according to the species is in a row 288 ways.
Pets can be grouped by permutation concept as follows.
Total number of pets = 10
Number of dogs = 4
Number of cats = 3
Number of alpacas = 2
Number of bunny = 1
The number of ways pets can be arranged in group according to the species in a row as,
⇒ (4!)(3!)(2!)(1!)
⇒ (24)(6)(2)(1)
⇒ 288
Hence we can conclude that the number of ways pets can be arranged in group according to the species in a row is 288 ways.
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What is the range of possible sizes for side x?
The range of the values are 8.3<x< 8.38
How to determine the range of valuesThe given figure is an isosceles triangle, with none of its side equal.
Using the Pythagorean theorem, we have
[tex] {x}^{2} = \sqrt{ {8}^{2} } + \sqrt{ {2.5}^{2} } [/tex]
[tex] x = \sqrt{70.25} [/tex]
[tex]x = 8.38[/tex]
Thus, the range of the values are 8.3<x< 8.38
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Each of the following is an example of quantitative data, except Group of answer choices weight in ounces income in dollars political affiliation age in years
The correct option is C. political affiliation.
An example of quantitative data are-
weight in ouncesage in yearsincome in dollarsWhat is quantitative data?Data that expresses a definite quantity, amount, or range is known as quantitative data. In most cases, the data is accompanied with measuring units, such as metres in the case of a person's height. Setting boundaries for such data makes sense, and performing mathematical operations on the data has significance.
Types of Quantitative Data are-
Counter: A measure of entities. For instance, the quantity of users who have downloaded a specific app from the App Store.Measurement of physical objects: Calculating the dimensions of any physical object is called measurement. The HR executive, for instance, meticulously measures the size of each cubicle given to the new hires.Sensory calculation: Mechanism to automatically "sense" the parameters being monitored in order to generate a continuous source of data. A digital camera, for instance, transforms electromagnetic signals into a string of numerical data.Data projection: Algorithms and other mathematical analysis techniques can be used to project future data. For instance, after conducting a thorough investigation, a marketer will forecast a rise in sales following the introduction of a new product.To know more about quantitative data, here
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The complete question is-
8. Each of the following is an example of quantitative data, except ______.
A. weight in ounces
B. income in dollars
C. political affiliation
D. age in years
The product of donnies age and 6 is 72. use variable d to represent donnies age
Answer: D times 6 = 72
D= 12
Step-by-step explanation:
The term product means multiplication. Donnie's age is represented by the letter d, so if the product of d and 6 is 72, or in other words, d times 6 is 72, then d would equal 12
Can someone help me with these problems and show work
Answer:
Khan Academy.
Step-by-step explanation:
You can go on khan academy and search up "percentages and numbers." The videos should immediately pop up. there are also lessons if you don't want to risk getting your problem wrong. If you are still confused and do not understand please just comment and I will respond.
The dot plots show the number of minutes that each of two sisters practiced the piano each day for 14 days.
ana
+++++
0 10
+
20
30
40
80
90
100
110
120
50 60 70
minutes practiced
tara
:
:
.
+++++ +
0 10 20
+
+
40
.
+ +
50 60 70 80
minutes practiced
++++
100 110 120
30
90
which sister showed more consistency in the amount of time she practiced? move a word to each blank to complete the sentence.
showed more consistency because the
of her data was less than that of her sister's data.
ana
tara
median
range
Averagely, Ana practiced for a greater amount of time each day.
What is the Mean and Median of a Data Set?Mean = sum of data points / number of data points.Median = center/middle of the data distribution.Mean for ana= (20 + 20 + 30 + 60 + 60 + 60 + 70 + 70 + 100 + 100 + 100 + 115 + 115 + 115)/14 = 69.6
Mean for tara= (20 + 20 + 30 + 30 + 45 + 45 + 45 + 70 + 70 + 70 + 75 + 90 + 90 + 120)/14 = 58.6
Median for ana= 70
Median for tara= (45 + 70)/2 = 57.5
Ana has the greater median and also has the greater mean.
Averagely, we can conclude that ana practiced for a greater amount of time each day.
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Find the value of a°,b°,x°,y° from the given figure
Answer:
Hope it helps u, any questions u may ask me
x=70 [alternate interior angles]
y=60 [alternate interior angles]
a=110 [By linear pair]
b=120 [By linear pair]
Suppose the expression a(b)n models the approximate number of people who registered for a dance program every day since the registration started, where a is the initial number of people who registered, b is the rate of increase in the number of people who registered every day, and n is the number of days since the registration started.
If the expression below models the registration for a particular dance program, what is the correct interpretation of the second factor?
There were 2.07 times as much people that applied in the fourth month than in the first month.
What is an exponential function?An exponential function is one that grows or decreases in an exponential manner. Thus, for any exponential function we can write; f(x) = ax^±n. The term a is the initial value x is the rate of change while n is the time elapsed. The positive sign is used for an exponential increase while the negative sign is used for an exponential decrease.
Thus we have;
f(x) = 27(1.2)^4
This shows us that the time elapsed is four months and f(x) = 2.07.
Therefore, there were 2.07 times as much people that applied in the fourth month than in the first month.
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50 POINTSS MORE !!!!!
6. The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block of ice to melt if the temperature was 45 degrees?
7. Ride Service. It costs $35 for a ride from the city center to the airport, 14 miles away.
a. Write the equation that relates the cost, c, with the number of miles, m.
b. What would it cost to travel 22 miles with this service?
The time it will take for the block of ice to melt if the temperature is 45 degrees is 3 hours.
The cost to travel 22 miles is 55 dollars.
How to find how long for the ice to melt?let
time it takes for ice to melt = x
air temperature = y
Therefore,
x ∝ 1 / y
x = k / y
when x = 2.5 hours and y = 54 degrees
Hence,
k = 2.5 × 54
k = 135
The time it will take for the block to melt at 45 degrees = 135 / 45 = 3 hours
How to find equation that relates the cost to the number of miles?It costs $35 for a ride from the city centre to the airport, 14 miles away.
where
c = cost
m = number of miles
Therefore, the equation is as follows:
35 / 14 = 2.5
c = 2.5m
Hence, when you travel 22 miles,
c = 22(2.5)
c = 55 dollars
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An isosceles triangle has two equal sides. If two of the sides are 5 inches and 2 inches, what is the perimeter of the triangle
Step-by-step explanation:
12 inches
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A restaurant uses rectangular napkins where the length, l, is twice as long as the width. the length of the napkin along the diagonal is x. what is x in terms of l? replace a and b with the correct values.
The value of x in terms of l is.
[tex]x=\sqrt{5l^{2} }[/tex]
What is Pythagoras Theorem?A fundamental relationship in Euclidean geometry between a right triangle's three sides is known as the Pythagorean theorem, or Pythagoras' theorem. The area of the square whose side is the hypotenuse is said to be equal to the total of the areas of the squares on the other two sides.
given data:length = [tex]l[/tex]
width= [tex]2l[/tex]
diagonal = [tex]x[/tex]
We would get a right angle triangle if we were to cut a part across the diagonal. The Pythagoras Theorem can be used to solve the right angle triangle.
[tex]x^{2} =a^{2} +b^{2}[/tex]
using this formula substituting the value:
[tex]x^{2} =l^{2} +2l^{2}[/tex]
[tex]x^{2} =l^{2} +4l^{2} \\x^{2} =5l^{2} \\x=\sqrt{5l^{2}} \\[/tex]
from the above calculation we have:
[tex]x=\sqrt{5l^{2} }[/tex]
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Which of the following is a simpler way to write StartFraction cosine theta Over sine theta EndFraction
The correct option is (d) cotФ.
As, cosineФ/sineФ = cotФ.
What are trigonometric identities?Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions.
There are numerous distinctive trigonometric identities that relate a triangle's side length and angle.
The basic trigonometric identities are-
Sine, (Sin) Cosine, (cos)Tangent, (tan)Secant, (sec)Cosecant (cosec)Cotangent (cot)Solution for the given trigonometric function; (cosineФ/sineФ)
Use formula of tanФ
tanФ = sineФ/cosineФ
Also, from basic trigonometric ratios.
tanФ = 1/cotФ
Substitute sineФ/cosineФ to get the result
1/cotФ = sineФ/cosineФ
Further simplify,
cotФ = cosineФ/sineФ
Therefore, the value of cosineФ/sineФ = cotФ
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Which of the following is a simpler way to write StartFraction cosine theta Over sine theta EndFraction-
(a) tanФ
(b) secФ
(c) cosecФ
(d) cotФ
Answer: D, cot o
Step-by-step explanation:
edge :-]
The triangle below is being reduced by a scale factor of two-thirds. What is the base of the new drawing? 15 m
Answer:
10
Step-by-step explanation:
A scale factor of 2/3 means multiplying all side lengths by 2/3
15*2/3=10
Are the triangles similar? If so, state the similarity and the postulate or theorem that justifies your answer.
Answer: no, it's not similar because we cannot multiply it by any factor that the ratio will be the
Step-by-step explanation:
In a bag there are 3 red 4 white, and 5 blue marbles. once a marble is selected it is not required. find: p(white, red) find: p(blue, white, red.)
Answer:
Below in bold.
Step-by-step explanation:
Total number of marbles = 3+4+5 = 12
P(white, red) = 4/12 * 3/11
= 12/132
= 1/11,
P(blue, white, red) = 5/12 * 4/11* 3/10
= 60/1320
= 6/132
= 1/22.
Which of the following numbers is a factor of 78?
.7
.4
.6
.9
PLEASE ANSWER AS SOON AS POSSIBLE!
Answer:
6
Step-by-step explanation:
If you divide 78 by all of the numbers, only one, 6, is a whole number, making it the only factor. You can check this by multiplying 6 by 13, (you get 13 from the division) and you get 78.
I hope this helps!
A binomial has two terms.
O True
O False
Answer: True.
Step-by-step explanation:
The prefix "bi-" shows that there are two terms in a binomial expression. Therefore, this statement is True.
I hope this helps! :)
Answer:
A. True.
Step-by-step explanation:
A binomial has two terms.
Think of it this way. Any object's name that starts with 'bi' has two of something. For example, a bicycle, bicycles have 2 wheels.
Hope this helps!
It cost Josiah $9.75 to send 65 text messages. How many text messages did he send if
he spent $28.20?
According to my calculation the answer is 188.
What is the length of the apothem, rounded to the nearest inch? recall that in a regular hexagon, the length of the radius is equal to the length of each side of the hexagon. 4 in. 5 in. 9 in. 11 in.
Answer:
Step-by-step explanation:
What is the length of the apothem, rounded to the nearest inch? recall that in a regular hexagon, the length of the radius is equal to the length of each side of the hexagon. 4 in. 5 in. 9 in. 11 in.
Answer:
9
Step-by-step explanation:
apothem (AB)
1/2 of the length (10) is 5
AB^2 + 5^2 = 10^
AB^2 = 100-25
Square root of AB^2 = square root of 75
AB = 8.66
8.66 is about 9
A new car is purchased for $45, 000 and over time its value depreciates by one half
every 3.5 years. What is the value of the car 17 years after it was purchased, to the
nearest hundred dollars?
How many pairwise comparisons must you make for a case with 15 statistically independent scenarios?
There are 105 comparisons made with 15 statistically independent scenarios.
According to the statement
we have given value of n is 15.
Then r = 2. so,
we use the formula NcR then
Substitute the values in it then
15c2
we open it according to ncr formula
In this formula we arrange the terms according to the following
NcR formula = n! /r! * (n-r)!
then the terms will arrange in this way.
So,
ncr = 15! / 2! * 13!
After solving the statement some terms will be cancel with each other and remaining terms are
ncr = 15*14/2
ncr = 210/2
ncr = 105
here the value becomes 105.
So, There are 105 comparisons made with 15 statistically independent scenarios.
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The compound periods multiplied by the number of years is 4t. The initial number of cars serviced is 920. The growth factor is represented by 1.03. The quarterly rate of growth is 0.03 or 3%. The growth rate is 1.03. 920(1.03) is the number of cars multiplied by 1.03. Base arrowRight Coefficient arrowRight Exponent arrowRight Rate
The number of compounding per year is[tex]N = 920(1+0.03)^4t[/tex]
How can the number of compounding per year be calculated?This can be calculated by making use of the formular; 920(1.03)^4t and use the compound interest equation which is N=P( 1+r/n)^nt
We know that 12% was given as the rate per year, then to get the quarter rate we divide by factor of 4, which is 12/4= 3%
The compounding for t years = n*t = 4t where n=4.
and the Initial number of cars serviced=920
Then we can substitute the values into the above equation, we have Thus [tex]N = 920(1+0.03)^4t[/tex]
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CHECK THE COMPLETE QUESTION BELOW:
Drag the tiles to the boxes to form correct pairs. Not all tiles will be used. A car repair center services 920 cars in 2012. The number of cars serviced increases quarterly at a rate of 12% per year after 2012. Create an exponential expression to model the number of cars serviced after t years. Then, match each part of the exponential expression to what it represents in the context of the situation. 920(1.03) is the number of cars multiplied by 1.03. The quarterly rate of growth is 0.03 or 3%. The compound periods multiplied by the number of years is 4t. The initial number of cars serviced is 920. The growth factor is represented by 1.03. The growth rate is 1.03. Exponent arrowRight Base arrowRight Coefficient arrowRight Rate arrowRight
Which two factors are needed to calculate the velocity of an object?
A. Speed
B. Direction
C. Mass
D. Acceleration
sutopa has 3/4 the money maneet has. maneet has $18 less than Kim. together, they have $286.40. How much money does each of them have?
The amount of money Kim, maneet and Sutopa have is $115.6, $97.6 and $73.2 respectively.
EquationKim = xManeet = x - 18Sutopa = 3/4(x-18)Total = $286.40x + (x - 18) + 3/4(x-18) = 286.40
x + x - 18 + 3/4x - 13.5 = 286.40
2 3/4x = 286.40 + 13.5 + 18
11/4x = 317.9
x = 317.9 ÷ 11/4
= 317.9 × 4 / 11
= 1,271.6 / 11
x = $115.6
So,
Kim = x
= $115.6
Maneet = x - 18
= 115.6 -18
= $97.6
Sutopa = 3/4(x-18)
= 3/4(97.6)
= $73.2
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Select the correct answer. if 6 cards are drawn at random from a standard deck of 52 cards, what is the probability that exactly 2 of the cards are spades? a. 0.038 b. 0.200 c. 0.315 d. 0.465 e. 0.747
Answer:
0.315
Step-by-step explanation:
Annette the shoe salesperson has two job offers. The Comfort over Style Shoe Emporium has offered her a $52,000 salary plus $3.25 for every pair of shoes she sells. The Heels for Fashion Slaves Boutique has offered a $37,000 salary plus $4.75 for every pair of shoes she sells. Over what range of shoe sales will Annette make more from the Comfort over Style job than she will from the Heels for Fashion Slaves job?
Using linear functions, it is found that over a range of x < 10000 she will make more from the Comfort over Style job than she will from the Heels for Fashion Slaves job
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.Considering the fixed salary as the intercept and the earning per shoe sold the slope, the functions for her salaries at each situation is:
C(x) = 52000 + 3.25xH(x) = 37000 + 4.75xHe will make more from the Comfort over Style job than she will from the Heels for Fashion Slaves job when:
C(x) > H(x)
52000 + 3.25x > 37000 + 4.75x
1.5x < 15000
x < 15000/1.5
x < 10000
The range is x < 10000.
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