Thus, the proof of the identity cos^2(5x) - sin^2(5x) = cos(10x) involves the use of the double angle formula for cosine. This identity is useful in solving various problems related to trigonometry.
To prove the trigonometric identity cos^2(5x) - sin^2(5x) = cos(10x), we will use the double angle formula for cosine.
This formula states that cos(2θ) = cos^2(θ) - sin^2(θ). We can rewrite our identity as:
cos^2(5x) - sin^2(5x) = cos(2 * 5x)
Using the double angle formula, we get:
cos^2(5x) - sin^2(5x) = cos(10x)
This proves the given trigonometric identity.
To understand this identity better, let's break it down.
The left-hand side of the identity consists of two terms, cos^2(5x) and sin^2(5x).
These terms are known as the Pythagorean identity and state that cos^2(θ) + sin^2(θ) = 1.
We can rewrite cos^2(5x) as 1 - sin^2(5x) using this identity.
Substituting this value in the given identity, we get:
1 - sin^2(5x) - sin^2(5x) = cos(10x)
Simplifying this equation, we get:
cos^2(5x) - sin^2(5x) = cos(10x)
Therefore, we have successfully proven the given trigonometric identity.
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y=x+3
y=-x-1
i need to find the solution of equations
Answer:
[tex]\huge\boxed{\sf \{(-2,1)\}}[/tex]
Step-by-step explanation:
Given equations:
y = x + 3
y = -x - 1
By comparing the above equations, we get:
x + 3 = -x - 1
Add x to both sides
x + x + 3 = -1 2x + 3 = -1Subtract 3 to both sides
2x = -1 - 3
2x = -4
Divide 2 to both sides
x = -4/2
x = -2Substitute x = -2 in first equation.
y = x + 3
y = -2 + 3
y = 1So,
Solution Set = {(x,y)}={(-2,1)}[tex]\rule[225]{225}{2}[/tex]
Answer:
[tex]x = - 2 \\ y = 1[/tex]
Step-by-step explanation:
The above are Simultaneous equations.
What are Simultaneous equations?Simultaneous equations are two or more algebraic equations which share same variables like X,Y or X,Y and Z.
They are called simultaneous equations because they can be solved the same time.
From the question,
[tex]y = x + 3 - - - - - - (1) \\ y = - x - 1 - - - - - - (2) \\ Substitute \: equation \: (2)into \: (1) \\ - x - 1 = x + 3 \\ Collect \: liketerms \\ - x - x = 3 + 1 \\ - 2x = 4 \\ Divide \: bothsides \: by \: - 2 \\ \frac{ - 2x}{ - 2} = \frac{4}{ - 2} \\ x = - 2 \\ Substitute \: the \: value \: of \: x \: into \: equation \: (1) \\ y = x + 3 \\ y = - 2 + 3 \\ y = 1 \\ Therefore \: x = - 2 \: \: and \: y = 1[/tex]
A pendulum is swinging next to a wall. the distance d(t) (in cm) between the of the pendulum and the wall as a function of time t (in seconds) can be modeled by a sinusoidal expression of the form a * sin (b * t) + d at t = 0, when the endulum is exactly in the middle of its swing, the bob is 5 cm away from the wall. the bob reaches the closest point to the wall, which is 3 cm from the wall, 1 second later. find d(t). (t should be in radians)
The equation of the function d(t) is d(t) = -2sin(π/2 t) + 5
How to determine the function d(t)?The function is given as:
d(t) = a * sin(b * t) + d
The bob is 5 cm away from the wall implies that the vertical shift is 5.
This means that
d = 5
So, we have
d(t) = a * sin(b * t) + 5
The bob reaches the closest point to the wall, which is 3 cm from the wall.
So, the amplitude is
A = 3 - 5
Evaluate
A= -2
So, we have:
d(t) = -2 * sin(b * t) + 5
The period is calculated as:
B = 2π/t
Where
t = (5 + 3)/2
This gives
t = 4
Substitute t = 4 in B = 2π/t
B = 2π/4
Evaluate
B = π/2
So, we have:
d(t) = -2sin(π/2 t) + 5
Hence, the equation of the function d(t) is d(t) = -2sin(π/2 t) + 5
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Let a and b be the events of the fda approving and rejecting a new drug to treat hypertension, respectively. The events a and b are:_____.
a. independent.
b. unilateral.
c. conditional.
d. mutually exclusive.
The correct option is (d) Mutually exclusive events.
Let a and b be the events of the fda approving and rejecting a new drug to treat hypertension, respectively. The events a and b are mutually exclusive events.
Two events are mutually exclusive or disjoint in logic and probability theory if they cannot both occur at the same time. The results of a single coin toss, which can result in either heads or tails but not both, provide a clear illustration.
A statistical term used to describe events that cannot occur concurrently is "mutually exclusive." It is frequently used to refer to circumstances in which the occurrence of one event takes precedence over the other. For instance, it is impossible for war and peace to exist together. They are therefore opposed to one another.
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4+5(7-1)+8/2
Simplify
Answer:38
Step-by-step explanation:4+5×6+4=4+30+4=38
Give the center and radius of a circle with the equation: (x+1)²+(y-2)²=100
A-The center is (-1. 2) and the radius is 10
B-The center is (1, -2) and the radius is 50
C- The center is (1, -2) and the radius is 10
D- The center is (-1, 2) and the radius is 50
OPTION A is the correct answer
From the top of a 16 m tall building, the angle of depression to a car on the road is 35°. To the nearest metre, how far is the car from the base of the building?
The distance of the car from the tall building to the nearest meter is 23m.
How far is the car from the base of the building?Given the data in the question;
Angle of depression θ = 35°Height of building = 16mDistance of car from building = xWe use trigonometric ratio since the scenario forms a right angle triangle.
tanθ = opposite / adjacent
tanθ = opposite / adjacent
tan( 35° ) = 16m / x
x = 16m / tan( 35° )
x = 16m / 0.7002
x = 23m
Therefore, the distance of the car from the tall building to the nearest meter is 23m.
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The Smith family bought carpet for their basement. It originally cost $1,600. The company offered and installment plan for 12 monthly payments of $150. How much was the carpet?
Answer:
$1,800
Step-by-step explanation:
If the Smiths agree to the installment plan and pay $150 every month for 12 months, it will add up to $1,800 at the end of the year.
Given the following table with selected values of f (x) and g(x), evaluate f (g(4)).
x –6 –4 1 3 4
f (x) 4 –1 –6 1 3
g(x) 1 4 3 –4 –6
–4
–1
1
4
The value of f(x) when x is -6 is 1.
Composite functionsComposite functions are also known as function of a function. In order to get a composite functions f(g(x)).
g(4) = -6 (the value of g(x) when x is equivalent to 4 is -6)
f(g(x))= f(g(4)) = f(-6)
f(-6) = 1
The value of f(x) when x is -6 is 1.
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What is the equation of the line that is used to show the reflection of a relation and its inverse?
help please and thank u <3
The graph and inverse of a function are reflections of each other along the line y = x.
A relation made up of ordered pairs (x, y) has as its inverse the set of all ordered pairs (y, x).
When a point is mirrored over the line y = x, the x- and y-coordinates swap places. Similarly, when a point is mirrored over the line y = -x, the x- and y-coordinates swap places and become negated.
Therefore,
The point (x, y)'s reflection over the line y = x is (y, x).
The point (x, y)'s reflection over the line y = - x is (-y, -x).
Thus, the graph and inverse of a function are reflections of each other along the line y = x.
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Which equation represents x, the side length of the greater square? x2 (x – 2)2 = 10 x2 2x2 = 10 x2 (x – 2)2 = 100 x2 2x2 = 100
Step-by-step explanation:
10x2(x-2)2=100x2x2 2x2
PLEASE HELP IM SUPER STUCK
Replace x with the value in the table and solve for y
-3: 2(-3)/3 + 4 = -6/3 + 4 = -2+4 = 2
0: 2(0)/3 +4 = 0/3 + 4 = 0+4 = 4
3: 2(3)/3 + 4 = 6/3 + 4 = 2 + 4 = 6
6: 2(6)/3 + 4 = 12/3 + 4 = 4+4 = 8
Answer:
When x = -3:
y = 2(-3)/3 + 4
y = 2
When x = 0:
y = 2(0)/3 + 4
y = 4
When x = 3:
y = 2(3)/3 + 4
y = 6
When x = 6:
y = 2(6)/3 + 4
y = 8
A machine in a candy company creates 9,328 pieces of candy each hour. if a box holds 98 pieces of candy, how many boxes can be filled in one hour?
Answer:
Step-by-step explanation:
9,328 and you would divide that by 98 and the answer would be 95.18367347 boxes
three decreased by eleven times a number
Step-by-step explanation:
3 - 11x
x is "a number".
so, the request is to multiply it by 11 and subtract the result from 3.
Rita divides avogadro’s number (approximately 6.02214 times. 1023) by 2.055 to calculate the number of atoms in a sample. which expression gives her result to the correct number of significant figures? 2.93 times. 1023 2.930 times. 1023 2.9305 times. 1023 2.93048 times. 1023
The correct answer is 2.930 × [tex]10^{23}[/tex]
What is Avogadro's number?
Avogadro's number, which is equal to 6.02214076 1023, is the quantity of units in one mole of any material (defined as its molecular weight in grams). Depending on the substance and the nature of the reaction, the units may be electrons, atoms, ions, or molecules (if any).
You have two numbers here. While number 2.0550 is stated with five significant figures, Avogadro's number is supplied with four. This implies that you must provide four significant figures with your result.
(6.022 × [tex]10^{23}[/tex]) / 2.0550 = 2.930 × [tex]10^{23}[/tex]
Therefore, the correct answer is 2.930 ×[tex]10^{23}[/tex] .
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given that 5x:9=7:3 calculate the value of x. give your answer in its simplest form.
[tex]\huge\underline{\red{A}\green{n}\blue{s}\purple{w}\pink{e}\orange{r} →}[/tex]
x = 21/5
Step-by-step explanation:
___________________________Given →5x:9 = 7:3
So,→ 5x/9 = 7/3
→ x = (7 × 9)/(3 × 5)
→ x = 21/5
___________________________Hope it helps you!!780tonnes 650kilograms ÷13
Answer:
60 kilograms
Step-by-step explanation:
help help help help help help pleaseeeeeee
the maximum height is 180ft.
How to find the maximum height of the rocket?The height of the equation is modeled by:
[tex]y = 60t - 5t^2[/tex]
Notice that this is a quadratic equation with a negative leading coefficient, which means that the maximum is at the vertex, then the maximum is at:
[tex]t = -60/(2*-5) = 6[/tex]
To get the maximum height, we need to evaluate the function in t = 6, we will get:
[tex]y = 60*6 - 5*6^2 = 180[/tex]
This means that the maximum height is 180ft.
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In the past, schools in Los Angeles County have closed an average of three days each year for weather emergencies. What is the probability that schools in Los Angeles County will close for four days next year
The probability that the schools in Los Angeles will be closed for four days next year is 0.1680.
What is Poisson's distribution formula?In order to comprehend separate events that happen at a steady pace over the course of a specified amount of time, Poisson's distributions are frequently utilized.
Formula for Poisson's distribution:
f(x) = ((λ^x)/x!).(e^(-λ))
x = Event to be calculated for
λ = Expected number of occurrences
P(X = x) = ((λ^x)/x!).(e^(-λ))
Here λ = 3 and x = 4
P(X = 4) = (81 / 4!).(e^(-3))
P(X = 4) = (81 * e^(-3))/24
P(X = 4) = (81 * 0.049)/24
P(X = 4) = 0.1680
Hence, the probability that schools in Los Angeles will be closed for four days next year is 0.1680.
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Green’s Sport Shop offers its salespeople an annual salary of $10,000 plus a 6% commission on all sales above $20,000. Every employee receives a Christmas bonus of $500. What are Mr. Cahn’s total earnings in a year in which his sales amounted to $160,000?
(A) $18,900
(B) $18,400
(C) $19,600
(D) $20,100
Mr. Cahn’s total earnings in a year is $18,900.
What is the total earnings?The total earnings is a function of the annual salary, commission and Christmas bonus.
Total earnings = annual salary + commission + Christmas bonus
Commission = 6% x (160,000 - 20,000)
6% x $140,000
0.06 x 140,000 = $8,400
Total earnings = $8,400 + 10,000 + $500 = $18,900
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A waste management survey collected data about the types of materials that were recycled by three different states in
2008
20082008. The table at left shows the approximate amount of each material recycled measured in thousands of tons. Based on the information in the table at left, as a percentage, approximately what is the relative frequency of plastics recycled in New Mexico in plastics recycled in all three states combined?
As a percentage, the relative frequency of plastics recycling in New Mexico in 2008 compared to plastics recycled in all three states combined is approximately 11%.
What is relative frequency?A relative frequency measures the proportion of events that occur within a class out of the total number of observations.
When using relative frequency, the results are usually displayed in percentages, proportions, and fractions.
Data and Calculations:Year State Recycled Plastics Relative
(thousands of tons) Frequency
2008 New Mexico 500 0.111 (500/4,500)
2008 New York 1,500 0.333 (1,500/4,500)
2008 Texas 2,500 0.566 (2,500/4,500)
Total 4,500 1
Thus, the relative frequency of plastics recycling in New Mexico in 2008 compared to plastics recycled in all three states combined is approximately 11%.
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Answer: 9%
Step-by-step explanation:
just did the question
Select one or more expressions that together represent all solutions to the equation.
Your answer should be in radians.
Assume n is any integer.
30 cos (30x) + 14 = -16
Choose all answers that apply:
Answer:
D. π/30 + n(π/15)
Step-by-step explanation:
The given equation can be solved by isolating the cosine function and using the inverse function to find the angles.
Solution30cos(30x) +14 = -16
30cos(30x) = -30 . . . . . . subtract 14
cos(30x) = -1
This is an extreme value of the cosine function that first occurs where the argument is π, and repeats every 2π after that:
30x = π +2nπ
x = π/30 +n(π/15) . . . . . odd multiples of π/30
__
Additional comment
The answer choices other than D include all multiples of π/30, or only even multiples of π/30. Either way, they are not part of the correct answer.
What is the ratio of the amount of the 30% solution to the amount of 5% solution used to create the 10% solution?
The ratio comes out to be 1:4.
What is ratio?
A ratio in mathematics shows how many times one number is contained in another. For instance, if a mixture of milk and water contains eight liters of water and six liters of milk, the ratio of water to milk is eight to six. Similarly, the ratio of water to the overall amount of mixture is 8:14, while the ratio of milk to water is 6:8.
Calculating the Ratio of the Solution:
Let us assume that x is the quantity of 30% of the solution and y be that of 5% solution, then,
30% of x + 5% of y = 10% of (x + y)
0.3x + 0.05y = 0.1 (x + y)
0.3x + 0.05y = 0.1x + 0.1y
0.3x - 0.1x = 0.1y - 0.05y
0.2x = 0.05y
Thus, the required ratio is,
0.2x / 0.05y = 1
x / y = 0.05 / 0.2
x / y = 5 / 20
x:y = 1 : 4
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On a hike, Maria walked up a hill in 30 minutes, and then she ran back down the hill in 10 minutes. Which of the following graphs could model the total distance Maria hiked?
The graph that describes the motion of Maria is the graph in Option B.
What is displacement?Displacement is the term that is used to describe the distance covered in a specified direction.
We are told that Maria walked up a hill in 30 minutes, and then she ran back down the hill in 10 minutes. The graph that describes this motion is the graph in Option B.
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Can anyone solve question 2 and 10 with step by step explaination??
Answer:
Question 2: 4 1/2
Question 10: $7.24
Step-by-step explanation:
Question 2:
1 1/2/x = 3/9
x: total pages read3: minutes first spent9: minutes spent total1 1/2: pages read in 3 minutesto solve for x, we need to use 1 1/2/x and multiply it by x:
1 1/2/x * x
then we need to have 3/9 multiplied by x:
3/9 * x
this is equal to 1 1/2 = 3/9 * x
now you need to multiply both sides by the opposite of 3/9 = 9/3
3/2 * 9/3 = 3/9 * 9/3 * x which equals 9/2 = 4 1/2
final answer: 4 1/2
Question 10:
Jeff bought 3 boxes of cereal $2.99 each which comes up to $8.97,
he bought one gallon of milk for $4.98,
and 4 bananas for $0.50 each which in total is $2.00.
total amount spent: 8.97 + 4.98 + 2.00 = $15.95
We know there is a 20% off discount so what we have to do is take 20% off $15.95!
20%= $3.19
15.95 - 3.19 = 12.76
The total is now $12.76. But, Jeff payed with a $20.00 bill. Because of this, he will get $7.24 back. 12.76 + 7.24 = 20
Final answer: $7.24
The name of this bronze-casting process relies on a modeled original form made from a pliable material. This method is known as ________.
The bronze-casting process that depends on a modelled original form is: lost-wax casting.
What is Lost-wax Casting?Lost-wax casting is a bronze-casting process which is also referred to as the cire-perdue. In this type of metal casting, a wax model is created when a molten metal is usually poured into a mold. The wax model is then melted and drained.
In summary, the name for this bronze-casting process that relies on this wax model is called: lost-wax casting.
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I really need help for my hw please
Answer:
5.46
111
86
367.2
49
467
use labels
Step-by-step explanation:
1. [tex]\frac{39}{2} times . 0.28[/tex]
[tex]\frac{39}{2} times \frac{28}{100} =\frac{1092}{200} \\[/tex]
[tex]\frac{1092}{200} = \frac{273}{50}[/tex]
Answer 273/50 or 5.46 inches
2. [tex]\frac{97}{87} times \frac{100}{1} =\frac{9700}{87}[/tex]
Answer 9700/87= 111%ft
3. [tex]\frac{146}{1} divided \frac{170}{1}[/tex]
[tex]\frac{146}{1}times\frac{1}{170} =\frac{146}{170}[/tex]
Answer=146/170=73/85=
86%tons
4. [tex]\frac{136}{1} times\frac{270}{100} =\frac{36720}{100}[/tex]
Answer=367.2%ft
5 .[tex]\frac{60}{123}[/tex]=49%inches
6.[tex]\frac{56}{1} times\frac{100}{12} =[/tex]467%inches
Given the equation, y=2x+6, what is its slope?
Select one:
O a. 6
Ob. 2
O c. 3
O d. 1
Answer:
B, 2
Step-by-step explanation:
The slope in a y=mx+b equation is m. The m in your equation is 2.
Therfore, your slope is 2.
Oscar bought some markers, notebooks, and packs of sticky notes for his office. He bought the same number of markers as notebooks. He bought three more packs of sticky notes than markers and notebooks combined. He paid $1.05 for each marker, $2.25 for each notebook, and the packs of sticky notes were $1.95. How many of each kind of office supply did Oscar buy if he paid $49.05 in all?
Using a system of equations, it is found that Oscar bought 6 markers, 6 notebooks and 15 packs of sticky notes.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the variables are:
Variable x: Number of markers.Variable y: Number of notebooks.Variable z: Number of packs of notes.He bought the same number of markers as notebooks, hence:
x = y.
He bought three more packs of sticky notes than markers and notebooks combined, hence:
z = 3 + x + y
z = 3 + 2x.
He paid $1.05 for each marker, $2.25 for each notebook, and the packs of sticky notes were $1.95. He spent $49.05 in total, hence:
1.05x + 2.25y + 1.95z = 49.05.
Then:
1.05x + 2.25x + 1.95(3 + 2x) = 49.05.
7.2x = 43.2
x = 43.2/7.2
x = 6.
The amounts are as follows:
y = 6.z = 3 + 2(6) = 15.He bought 6 markers, 6 notebooks and 15 packs of sticky notes.
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Which factorization could represent the number of water bottles and weight of each water bottle?
The factorization that could represent the number of water bottles and weight of each water bottle is 12(5x^2 + 4x + 2). Option B
What is factorization?The term factorization has to do with the process of obtaining common factors in an expression. It involves dividing each term in the expression with a factor that is common to all the terms in the expression.
The factorization that could represent the number of water bottles and weight of each water bottle is 12(5x^2 + 4x + 2).
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Missing parts;
Mara carried water bottles to the field to share with her team at halftime. The water bottles weighed a total of 60x2 + 48x + 24 ounces. Which factorization could represent the number of water bottles and weight of each water bottle? 6(10x2 + 8x + 2) 12(5x2 + 4x + 2) 6x(10x2 + 8x + 2) 12x(5x2 + 4x + 2)
(?) James is following this recipe to make flapjacks.
James uses 650 g of margarine.
How many flapjacks is James making?
Recipe: Makes 8 flapjacks
50 g sugar
100 ml syrup
125 g oats
130 g margarine
65 g sultanas