The simplified form of the equation cos(θ) cos(2θ) + sin(θ) sin(2θ) = √2/ 2 is 4 cos³θ − 3 cos θ − √2/2 = 0.
The equation to use an addition or subtraction formula to simplify is given as:
cos(θ) cos(2θ) + sin(θ) sin(2θ) = √2/ 2
We know that cos 2θ = 2cos²θ − 1 and sin 2θ = 2sinθ cosθ.
Replacing these values in the above equation, we get:
cos θ (2 cos²θ − 1) + sin θ (2 sin θ cos θ) = √2/2
Simplifying the above equation, we get:
2 cos²θ cos θ − cos θ + 2 sin²θ cos θ = √2/2
Using the identity cos²θ + sin²θ = 1, we can substitute cos²θ = 1 − sin²θ in the above equation to get:
2 cos θ (1 − sin²θ) − cos θ + 2 sin²θ cos θ = √2/2
Simplifying further, we get:
2 cos θ − 2 cos³θ − cos θ + 2 sin²θ cos θ = √2/2
Rearranging and simplifying, we get:
(2 cos θ − cos θ − √2/2) + (2 cos³θ − 2 sin²θ cos θ) = 0
Using the identity sin²θ + cos²θ = 1, we can substitute sin²θ = 1 − cos²θ in the second term of the above equation to get:
(2 cos θ − cos θ − √2/2) + (2 cos³θ − 2 cos θ + 2 cos³θ) = 0
Simplifying, we get:
4 cos³θ − 3 cos θ − √2/2 = 0
Now, we can solve this cubic equation using a numerical method like the Newton-Raphson method to get the value of θ that satisfies the given equation.
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A square pyramid is shown. What is the surface area?
A square based pyramid, with bases labeled 3.5 centimeters and side length of triangle labeled 7 centimeters.
(4 points)
30.625 cm2
61.25 cm2
19.25 cm2
49.625 cm2
The surface area of the square pyramid is B)61.25[tex]cm^2[/tex]
What is surface area of square pyramid?
The term "square pyramid" also refers to a polyhedron having a square base and four triangular faces. Where the four faces come together is known as the apex. The pyramid's faces connect the base and the tip. The calculation for the square pyramid's surface area adds the areas of the four triangular faces and the base.
Here the given measures are base = 3.5 cm and height = 7cm.
Now using surface area formula then
Surface area of the square pyramid = Base area + 2bl square unit.
Base area = 3.5 × 3.5 = 12.25[tex]cm^2[/tex]
=> Surface area = 12.25+ 2×3.5×7
=> Surface area = 12.25+49
=> Surface area = 61.25[tex]cm^2[/tex]
Hence the surface area of the square pyramid is B)61.25[tex]cm^2[/tex]
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The graph of g(x) = (One-half)x is the graph of f(x) = 2x reflected over the y-axis. Which graph represents g(x)?
Graph of g(x) = (1/2)ˣ i.e. Graph of f(x) = 2x reflected over the y-axis is drawn below.
What is function?
A function is a process or a relation that associates each element 'a' of a non-empty set A , at least to a single element 'b' of another non-empty set B. A relation f from a set A (the domain of the function) to another set B (the co-domain of the function) is called a function in math.
f = {(a, b)| for all a ∈ A, b ∈ B}.
Given, the function
f(x) = (2)ˣ
g(x) = (1/2)ˣ
g(x) is the graph of f(x) reflected over y- axis.
The rule of reflection over y-axis:
(x, y) → (-x, y)
Thus, when a graph gets reflected over y-axis it means that horizontal components are additive inverse of the original horizontal components.
Therefore the graph of g(x) = (1/2)ˣ is as follows.
Hence, below is the graph of g(x) = (1/2)ˣ that is graph of f(x) reflected over y- axis.
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Rashid travels 240km from Abu Dhabi to Sharjah driving 120km/h for 2 hours. His parents strongly suggest he slow down by v km/h, even though it will take him t hours longer to make the trip, or he will lose his driving privilege. Which relation shows time, t, it will now take Rashid to make the trip relative to the speed, v with which he slows down?
Answer: The relation between the time it will take Rashid to make the trip, t, and the speed at which he slows down, v, can be determined by using the formula:
distance = speed * time
We know that Rashid travels 240 km from Abu Dhabi to Sharjah and that he originally drove at 120 km/h for 2 hours. So, we can write the equation:
240 = 120 * 2
Now, if Rashid slows down by v km/h, then the new speed is (120 - v) km/h and the new time, t, can be determined by substituting the new speed and distance into the equation:
240 = (120 - v) * t
So the relation shows that t = (240 / (120 - v))
This relation shows that the time, t, it will now take Rashid to make the trip is directly proportional to the speed, v, at which he slows down. As Rashid slows down, the time taken to make the trip will increase, and as he speeds up, the time will decrease.
Step-by-step explanation:
What is 4/3 as an exponent?
Answer:
4/3 as an exponent is 4 to the 3rd power ( so four with a three right above it a little)
Step-by-step explanation:
One solution, no solutions, or infinitely many solutions? Write another system of equations with the same number of solutions that uses the first equation only.
12x+51y=156
-8x-34y=-104
The system of equations 12x + 51y = 156 and -8x - 34y = -104 has infinitely many solutions
What is an equation?An equation is an expression composed of variables and numbers linked together by mathematical operations.
Given the equation:
12x + 51y = 156
Dividing by 3:
4x + 17y = 52 (1)
Also:
-8x - 34y = -104
Dividing by 2:
-4x - 17y = -52 (2)
To solve both equation by elimination, add equation 1 and 2:
0 = 0
The equation has infinitely many solutions
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ann and byron positioned themselves 35m apart on one side of a stream
Ann and Byron set up shop 35 meters apart on one side of a creek, with the height of the cliff on the other side of the stream being 106.73 meters.
what is triangle ?Since a triangle has three sides and three vertices, it is a polygon. It is a fundamental geometric shape. Triangle ABC is the moniker given to a triangle that has vertices A, B, and C. When the three points are not collinear, a singular plane and triangle are found in Euclidean geometry. A triangle is a polygon if it has three sides and three corners. The points where the three sides meet are known as the triangle's corners.
given
consider the triangles ABC and ACD
from the figure
∠DAC = 36°
∠CAB = 68°
AD = 35 m
we have to find the height of the cliff i.e. he length BC
i.e. the length BC
cosθ = adjacent side / hypotenuse
cos36° = 35 / AC
AC = 43.21 m
tan68° = BC / 43.21
BC = 43.21 * 2.47 = 106.73 m
Ann and Byron set up shop 35 meters apart on one side of a creek, with the height of the cliff on the other side of the stream being 106.73 meters.
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A cyclist rode for 3.5 hours and completed a distance of 60.9 miles. If she kept the same average speed for each hour, how far did she ride in 1 hour?
Part A
Estimate the quotient. Include an equation to show your work. Explain your thinking.
Part B
Find the exact distance she rode in 1 hour by completing the division. Enter the answer in the box.
Miles:
A. The quotient is given as follows: 60.9 miles / 3.5 hours.
B. The exact distance that she rode in one hour is of: 17.4 miles.
How to obtain the distance?The measures in this problem are obtained considering the relation between velocity, distance and time.
The relation between velocity, distance and time states that the velocity is obtained by the quotient of the distance by the time, as follows:
v = d/t.
In which:
v is the velocity.d is the distance.t is the time.The parameters for this problem are obtained as follows:
Distance of 60.9 miles.Time of 3.5 hours.Hence the average velocity is obtained as follows:
v = d/t
v = 60.9/3.5
v = 17.4 miles per hour.
This means that the exact distance that she rode in one hour is obtained as follows:
17.4 = d/1
d = 17.4 miles.
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In the past year, Christine watched 44 movies that she thought were very good. She watched 55 movies over the whole year. Of the movies she watched, what percentage did she think were very good?
Answer:
80 %
Step-by-step explanation:
Christine enjoyed 44 out of the 55 movies she watched.
This can be represented as
44/55 movies. This fraction simplified is 4/5 or 80 %
In rectangular park that is 63 yards by 16 yards, a dog began in the northwest corner and ran south along the length of the park. Then the dog ran east along the width to the southeast corner. Finally, the dog ran back to the northwest corner. How far did the dog run?
If in rectangular park that is 63 yards by 16 yards, a dog began in the northwest corner and ran south along the length of the park and Finally, the dog ran back to the northwest corner. The dog ran 65 yards.
What is distance?Distance can be defined as how far an object is from another object.
Using the Pythagoreans theorem to find the distance
c² = √a² + b²
Let plug in the formula
c² = √63² + 16²
c² = √3,969 + 256
c = √4,225
c = 65 yards
Therefore the distance is 65 yards.
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Which could be the measures of the three angles of an acute triangle.A. 40, 90, 50B. 45, 45, 90C. 25, 25, 130D. 70, 80, 30
Answer:
Option D
Step-by-step explanation:
Remember that the sum of all angles in ANY triangle is 180 degrees. Also, remember that an acute triangle has angles that are all less than 90 degrees. So not only do our options have to add up to 180 degrees, but they also have to be less than 90 degrees in order for the triangle to be acute.
We can immediately exclude options A, B, and C because all three options have angles greater than 90 degrees, which means the triangle isn't acute.
Your answer is option D or "70, 80, 30." Since...
70 + 80 + 30 = 180
70 < 90
80 < 90
30 < 90
Hope this helps.
Find an equation for the perpendicular bisector of the line segment whose endpoints are (−2,−9) and (−6,7)
Answer: y = 1/4x +1
Step-by-step explanation:
First, we can find the midpoint of the line segment using the average of x and y coordinates of the endpoints:
M(x,y) = ((-2 + (-6))/2 , (-9 + 7)/2)
M(x,y) = (-4, -1)
To find the slope of the perpendicular bisector, we have to take the negative reciprocal of the slope of the line segment which is (y2-y1)/(x2-x1) = (7-(-9))/(-6-(-2)) = -16/4 = -4
Slope of the perpendicular bisector = -1/slope of the line segment = -1/(-4) = 1/4
Now we can use the point-slope form to find the equation of the line:
y - y1 = m(x - x1)
y - (-1) = 1/4(x + 4)
y = 1/4x +1
so the equation of the perpendicular bisector of the line segment whose endpoints are (-2,-9) and (-6,7) is y = 1/4x +1
Which graph represents the function
f (x) = |x+2| +1?
Answer: The first graph
Step-by-step explanation:
Graph the absolute value using the vertex and a few selected points.
Hope this helps !
Find the exact volume of a waffle ice cream cone with a 3-in. diameter and a
height of 15 inches.
Answer:
Step-by-step explanation: v = 1/3π[tex]r^{2}[/tex]h
[tex]\frac{1}{3}[/tex]π([tex]1.5^{2}[/tex]) x 15
[tex]\frac{1}{3}[/tex]π x 2.25 x 15
=35.34291
Suppose f(x) is a function such that if p
Of(x) can be odd or even.
Of(x) can be odd but cannot be even.
Of(x) can be even but cannot be odd.
Of(x) cannot be odd or even.
Answer:
Step-by-step explanation:
Which of the following statement is correct regarding Sampling Distributions?a. The sampling distribution of the sample mean will be approximately normal.b. The mean of the sampling distribution of the sample mean will be equal to the population mean.c. Both A and B.d. None of the above.
Hence option b is the correct option i.e The sample mean's sampling distribution will be roughly normal.
The appropriate response to the question about Sampling Distributions is The sample mean will have a sampling distribution mean that is identical to the population mean.
The sampling distribution must then resemble the normal distribution as the sample size grows.
According to this, if the sample size is large enough, the sampling distribution of the mean will always be normally distributed. The sampling distribution of the mean will be normal regardless of whether the population has a normal, Poisson, binomial, or any other distribution.
A bell-shaped, symmetrical distribution known as a normal distribution has fewer observations the farther out from its center.
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Samuel pays $59.99 plus $0.25 for each minute over 450 minutes for his cell phone plan. Monica pays $64.99 plus $0.10 for each minute over 450 minutes for her cell phone plan. Which system of equations represents this situation? Let x represent the number of minutes over 450.
Using the slope and y - intercept, the linear function that models this problem is y = -33.333x + 68.32
What is System of Linear EquationsA system of linear equation is a set of at least two linear equations containing the same set of variables. Linear equations use only polynomial coefficients and the operations of addition, subtraction, multiplication and division. The solutions to a linear equation system are the values of the variables which make all the equations true.
To solve this problem, we have to define our variables and set our system of linear equations.
Let;
x = number of minutesy = total costUsing the data given, we can find the slope and y - intercept of a linear function that represents this.
slope (m) = y₂ - y₁ / x₂ - x₁
m = 64.99 - 59.99 / 0.10 - 0.25
m = -33.333
Using the slope in the slope-intercept form of a linear equation
y = mx + c
Taking one point;
59.99 = -33.333(0.25) + c
c = 68.32
The linear equation can be written as;
y = -33.33x + 68.32
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Solve the following system of equations graphically on the set of axes below.
Y = -1/6x+7
Y = 2x-6
The solution for the given system of equations would be (6, 6).
What is the system of equations?
A system of equations is a set of two or more equations that are solved simultaneously. Each equation in the system contains one or more variables, and the goal is to find the values of the variables that satisfy all the equations in the system.
y = -1/6x + 7
y = 2x - 6
To graph the equations y = -1/6x + 7 and y = 2x - 6, we can follow these steps:
Plot the y-intercepts of the equations: The y-intercept is the point where the line crosses the y-axis. For the equation y = -1/6x + 7, the y-intercept is (0,7), and for the equation y = 2x - 6, the y-intercept is (0,-6).
Draw the lines: Starting from the y-intercepts, use the slope of the lines to draw the lines. The slope of the first equation is -1/6 and the slope of the second equation is 2.
Find the x-intercept: The x-intercept is the point where the line crosses the x-axis. For the equation y = -1/6x + 7, the x-intercept is (-42,0) and for the equation y = 2x - 6, the x-intercept is (-3,0)
Locate the point of intersection: The point of intersection of the two lines is the solution of the system of equations. The point of intersection for the given equation is (3,1).
Label the axes and the lines: Label the x- and y-axes, and give the lines descriptive names or equations.
Now draw the graph, we get
Hence, the solution for the given system of equations would be (6, 6).
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Segment BD bisects segment AC. Solve for x. Round to the nearest tenth, if necessary. (Image not necessarily to scale.)
Given that segment BD bisects segment AC, the value of x is 5.8 units.
Properties of similar triangles.If the corresponding sides and/ or internal angles of two shapes have some common properties, then it can be said that the shapes are similar. Though similarity does not mean congruency.
Given that segment BD bisects segment AC, the value of x can be determined by comparing the sides of triangles ABD and BCD.
Thus, we have;
x/ 7 = 14/ 17
17x = 7 * 14
= 98
17x = 98
x = 98/ 17
= 5.76
x = 5.8 units
Thus to the nearest tenth, the value of x is 5.8 units.
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Which equation can be used to find what percent 18 is of 125
Answer: 22.5
Step-by-step explanation: To get a percentage of a number you multiply the number you want the percentage of (125), and multiply it by your percent in decimal form. So basically the equation went like this: 125 x 0.18 = 22.5. Another example of this would be if you wanted to find 60 percent of 20. You would just take 60 and put it in decimal form and multiply. 20 x 0.60 = 12. Hope this helps!
solve the system. 3x + y = 10 y = 6x + 1
17 7/9 divided by 0. Please hurry I have to get this done soon. Thank uu :D
Answer:undefined
Step-by-step explanation:
Which relation has an inverse function?
An inverse function or anti function is defined as a function, in which the graph can be reverse into another function. hence commonly , if any type of function “f” takes x to y then, the inverse of “f” function will take y to x. hence the function is represented as ‘f’ or ‘F’, then the inverse function is denoted by [tex]f^{-1} and F^{-1}[/tex].
let us assume that f and g are inverse functions, then the condition for f(x) = y if and only if g(y) = x
when the graph is represented in an original function which has to be one-to-one function in order to assure that its inverse is an function. A function is said to be a one to one function only if function of every second element corresponds to the first value .
The graph of function is the inverse of a function which reflects two things, that is , one represents the function and second represents the inverse of the function, over the line y = x. This line in the graph passes through the origin and which has slope value equal to 1 and represented as;
y = f-1(x) which is equal to x = f(y)
This relation is basically similar to y = f(x), which tells that graph of f but the part of x and y are reversed here. So if we have to draw the graph of f-1, then we have to manipulate the positions of x and y in coordinate axes.
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What is 7t + 2 = -12
Determine the y-intercept of -5x+7y=8
Answer:
The y-intercept of -5x+7y=8 is (0,8/7)
Step-by-step explanation:
The y-intercept of a linear equation is the point where the graph of the equation crosses the y-axis. In this equation, when x = 0, the equation becomes 7y = 8, so the y-intercept is (0, 8/7).
Answer:I hope this helps
Step-by-step explanation:
-5x+7y=8
To find the -intercept, substitute x=0 something like this
-5 times 0+7y=8
Solve the equation for Y which is something like
y=8/7 this your answer for the y intercept.
To find the quotient of 14 and 8, consider that some number times 8 equals 14. Since 14 is equivalent to Response area, that number must be Response area. Then 14÷8=Response area, because of the inverse relationship between multiplication and division.
Since 14 is equivalent to 8x, that number must be 14 ÷ 8. Then 14 ÷ 8 = x, because of the inverse relationship between multiplication and division.
How to use inverse relationship of division?There are 4 basic mathematical operations which are addition, subtraction, division and multiplication.
Multiplication and division are inverse operations because multiplication undoes division and also division undoes multiplication.
We want to find the quotient of 14 and 8. This is expressed as 14 ÷ 8.
Now, we are told to consider that some number times 8 equals 14. Let the number be x and as such we have;
8x = 14
Now, since 14 is equal to 8x, then that number must be gotten by using the inverse relationship by multiplying both sides by 1/8 to get;
x = 14/8
Finally, 14÷8 = x because of the inverse relationship between multiplication and division.
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What is sum of angles of a triangle class 7?
According to the triangle's "angle sum property," a triangle's interior angles add up to 180°.
The smallest polygon with three sides and three interior angles is a triangle.
A triangle is a three-sided polygon that consists of three vertices, three sides, and three angles. Three types of triangles—scalene, isosceles, and equilateral—can be distinguished based on the length of their sides.
It can be an acute-angled, obtuse-angled, or right-angled triangle depending on the size of its angles. A triangle's internal angles add up to 180 degrees. Two more terms—altitude and triangle median—are introduced to you in this essay.
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the admision to a local fair is $10 each for adult and $6 for each child. Each ride costs $1.50 for an adult and $1 for a child.
An adult and child will each go on 7 ride. How much more did the adult spend?
The cost of admission for an adult is $10 and for a child is $6
The cost of each ride for an adult is $1.50 and for a child is $1
The adult will spend $10 for admission + $1.5 x 7 rides = $17.5
The child will spend $6 for admission + $1 x 7 rides = $13
The adult spent $17.5 - $13 = $4.5 more than the child.
Can the sides of a triangle have lengths 4 7 and 8?
It is possible to construct a triangle with lengths of its sides 8cm, 7cm and 4cm because the sum of two sides of a triangle is greater than the third side.
Now, According to the question:
We know that:
What is triangle inequality rule?
The triangle inequality theorem states that for any given triangle, the sum of the two sides is always greater than the third side. The Triangle is a polygon with three line segments as its boundaries.
Suppose A, B and C are three sides of a triangle, then by using triangle inequality rule, A + B > C then A, B and C make a triangle.
We have the sides:
4 , 7 and 8
4 + 7 > 8
It is possible to construct a triangle with lengths of its sides 8cm, 7cm and 4cm because the sum of two sides of a triangle is greater than the third side.
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p:q=2/3:5/6 and q:r=3/4:1:2,find p:q:r
Answer:
p:q:r = 12:15:10
Step-by-step explanation:
Given p:q = (2/3):(5/6) and q:r = (3/4):(1/2), you want p:q:r.
Integer ratiosWe can turn each ratio into a ratio of integers by multiplying by the least common denominator.
p:q = 4:5 . . . . . . . multiply the given ratio by 6
q:r = 3:2 . . . . . . . multiply the given ratio by 4
Now, we need to have q be represented in each ratio by the same number. That number will be the least common multiple of its representations in these ratios: LCM(5, 3) = 15.
Multiplying the first ratio by 3, we have ...
p : q = 12 : 15
Multiplying the second ratio by 5 gives ...
q : r = 15 : 10
Ratios of all threeNow, we can write the ratio of interest:
p : q : r = 12 : 15 : 10
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please what is 4 times 5
_
12
Answer:20
Step-by-step explanation: 5+5+5+5
Answer:
4 times 5 is equal to 20.
Step-by-step explanation: