3. Show how the half angle formula for sin A/2 can be used to find the exact value of sin (202.5°). Show all work (including reference triangles) to simplify your answer.
Answers
To use the half angle formula for sin A/2 to find the exact value of sin (202.5°), we first need to write 202.5° as a sum or difference of two angles whose sine values we know.
Since 202.5° is not a standard angle, we can use the fact that it is halfway between 180° and 225° to write:
sin (202.5°) = sin [(180° + 225°)/2]
Now we can apply the half angle formula for sin A/2:
sin (A/2) = ± √[(1 - cos A)/2]
where the sign depends on the quadrant of A/2. In this case, A = (180° + 225°)/2 = 202.5°, which is in the third quadrant where sine is negative. Therefore, we take the negative sign:
sin (202.5°) = -√[(1 - cos 405°)/2]
To find the value of cos 405°, we can use the fact that cos (360° + α) = cos α for any angle α. Therefore:
cos 405° = cos (360° + 45°) = cos 45° = √2/2
Substituting this into our formula for sin (202.5°), we get:
sin (202.5°) = -√[(1 - √2/2)/2]
To simplify this expression, we need to rationalize the denominator. Multiplying numerator and denominator by √2, we get:
sin (202.5°) = -√[2 - √2]/2
Now we can draw a reference triangle in the third quadrant with angle A/2 = 101.25°, opposite side √[2 - √2]/2, and hypotenuse 1. Using the Pythagorean theorem, we can find the adjacent side:
a² + (√[2 - √2]/2)² = 1²
a² = 1 - (2 - √2)/4 = (2 + √2)/4
a = ± √[(2 + √2)/4]
Since cosine is negative in the third quadrant, we take the negative sign:
a = -√[(2 + √2)/4]
Therefore, the exact value of sin (202.5°) is:
sin (202.5°) = -√[2 - √2]/2 = -a = √[(2 + √2)/4]
To find the exact value of sin(202.5°) using the half angle formula, we'll need to use the following formula:
sin(A/2) = ±√[(1 - cos(A))/2]
In this case, A = 202.5° x 2 = 405°. Now, we need to find the reference angle for 405°. Since it's 45° more than a complete circle (360°), the reference angle is 45°.
Next, we need to find the cos(45°). Using the 45-45-90 reference triangle, we know that cos(45°) = √2/2.
Now, we can plug this into the half angle formula:
sin(202.5°) = ±√[(1 - cos(405°))/2] = ±√[(1 - (√2/2))/2]
Since 202.5° is in the third quadrant where both sine and cosine are negative, we'll take the negative root:
sin(202.5°) = -√[(1 - (√2/2))/2]
Now, simplify the expression:
sin(202.5°) = -√[(2 - √2)/4]
This is the exact value of sin(202.5°).
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Related Questions
Write an equivalent expression in expanded form. 9 (4x+3y2/3)
Answers
Answer:
36x + 27y + 6
Jessica paid $23,000 for her car. The value of the car depreciates at a rate of 15% each year. Which function represents the value of her car each year?
Answers
Answer:
Step-by-step explanation:
DOES ANYONE KNOW THE ANSWER
The solution to the system is (6,2). Select all the equations that could be the other
equation in the
system.
a.y = -3x
C. y=x+6
b.y=-*-* +3
d. y = {x-1
e. y=1&x-1
f.y = 4x - 2
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Answer:
Step-by-step explanation: 645
Answer: C and E
Step-by-step explanation:
i took this on my quiz and got it right :)
describe and correct the error in using properties of parallelograms.
Answers
Error: The error in using properties of parallelograms is assuming that all quadrilaterals with opposite sides that are parallel are parallelograms.
Correction: Not all quadrilaterals with opposite sides that are parallel are parallelograms. To determine if a quadrilateral is a parallelogram, both pairs of opposite sides must be parallel and equal in length. Additionally, the opposite angles must be congruent. These conditions must be satisfied for a quadrilateral to be classified as a parallelogram.
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A circle is the set of all points that are the same distance from one given point. Find an example that contradicts this definition
Answers
A circle is a curve that is made by a moving point in a plane. The perimeter of a circle is equal to the length of the circle's defining line and is commonly referred to as the circle's circumference.
What is a circle?
A circle can be thought of as a curve drawn out by a point traveling in a plane so that its distance from a particular point remains constant.It can also be thought of as the shape made by all of the points in a plane that are at a specific distance from the center.The claim that the provided point needs to be in the center of the circle contradicts this interpretation of the circle.The perimeter of a circle serves as an example of such an ambiguous term. Often referred to as the circumference of a circle, it is defined as the length of the line encircling it.To learn more about circles, refer to the link: https://brainly.com/question/25938130
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.role perception indicates our view of how we are supposed to act in a given situation. TRUE OR FALSE?
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True. Role perception refers to an individual's perception of what their role or position entails and how they are expected to behave in a particular situation.
It is shaped by various factors such as the individual's personal values, beliefs, experiences, and cultural background, as well as the expectations and norms of the organization or group they belong to.
Role perception can influence an individual's behavior, performance, and job satisfaction, as it guides them on how to act and what to expect from others in their role. However, role perception may not always be accurate or aligned with the actual expectations of the role, which can lead to misunderstandings and conflicts.
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Help step by step plz :)
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Answer:
182i2j3jdii32i3i38839393838383383 yan yung sagot pasalamat ka nalang mamaya
Step-by-step explanation:
hakdog ses dog
6x=54-12
6x=42
6x/6=42/6(divide both sides)
x=7
Consider the problem Min3X2−22X+2XY+y2−16Y+60 s.t. X+5Y≤8 a. Find the minimum solution to this problem. If required, round your answers to two decimal places. The optimal solution is X=Y= for an optimal solution value of b. If the right-hand side of the constraint is increased from 8 to 9 , how much do you expect the objective function to change? If required, round your answer to two decimal places. The optimal objective function value will by c. Re-solve the problem with a new right-hand side of 9 . How does the actual change compare with your estimate? If required, round your answers to two decimal places. The new optimal objective function value is so the actual is
Answers
(a) The given problem can be written as:
Minimize: \(3X^2 - 22X + 2XY + Y^2 - 16Y + 60\)
(b) Subject to: \(X + 5Y ≤ 8\)
(c) We can find the difference between the new optimal objective function value and our estimated change.
a. To find the minimum solution to the given problem, we need to solve the optimization problem by minimizing the objective function subject to the constraint.
The given problem can be written as:
Minimize: \(3X^2 - 22X + 2XY + Y^2 - 16Y + 60\)
Subject to: \(X + 5Y ≤ 8\)
To find the minimum solution, we need to solve this problem using optimization techniques like linear programming or calculus.
b. If the right-hand side of the constraint is increased from 8 to 9, we need to analyze the change in the objective function.
To do that, we can find the difference between the objective function value at the new solution and the old solution.
c. Re-solving the problem with a new right-hand side of 9 will give us a new optimal objective function value.
To compare the actual change with our estimate, we can find the difference between the new optimal objective function value and our estimated change.
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in rectangle QRST , QS = 5x-31 and RT= 2x+11. Find the lengths of the diagonals of QRST
Answers
Answer:
11.7
Step-by-step explanation:
I just solved and added them all-
⚠️HELP PLZZZ ⚠️‼️help plz I’ve been on this question for an while now and I don’t understand
Answers
Answer:
3:6
Step-by-step explanation:
8:24 and 24:72 reduce down to 1:3. 3:6 is 1:2.
your calculations indicate that you will need 20.8 cubic yards of concrete for a slab floor. which of the following quantities would be the best amount to order accounting for waste?
Answers
Answer:
Step-by-step explanation:Your calculations indicate that you will need 20.8 cubic yards of concrete for a slab floor. Which of the following quantities would be the best amount to order accounting for waste
assume that a sample is used to estimate a population proportion p. find the margin of error m.e. that corresponds to a sample of size 306 with 79.1% successes at a confidence level of 99.5%.m.e.
Answers
The margin of error for the statistical scenario described is 0.0599
To obtain the margin of error , we use the formula:
ME = z * √(p*(1-p)/n)p = 0.7911 - p = 0.209n = 306Zcrit at 99.5% confidence interval = 2.576Inserting the formula as follows:
ME = 2.576 * √(0.791 * (0.209)/306)
ME = 2.576 * 0.0232
ME = 0.0599
Therefore, the margin of error is 0.0599
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if joan reads 96 pages in 4 hours how long will it take her to read 240
Answers
Answer:
10 hours to readm240
Step-by-step explanation:
Answer:
10 hours.
Step-by-step explanation:
If you divide 96 by 4, you get 24. Then divide 240 by 24, and you get 10.
WILL MARK brainiest answer
Answers
Answer:
i Think it would be B
Step-by-step explanation:
. Two trains leave a train station traveling different directions. The first train travels 12 miles west, then 6 miles north. The second train travels 20 miles east, then 35 miles north. a. The train station is the origin. What is the coordinate of each train? b. Using the train station and the stop point of the first train, what is the slope of the line? Is it horizontal, vertical or neither? Write the equation of the line in slope-intercept form and standard form. c. The city wants to build a second train station 2 miles directly north of the first train station. They are going to build a train track from the second train station that is parallel to the path the first train would've traveled if it had taken a direct route to its stopping point. What would be the equation of the line in slope intercept form of the new track?
Answers
Answer:
a. The coordinates of the first train are (-12,6). The coordinates of the second train are (20,35).
b. The slope of the line for the first train is \(-\frac{1}{2}\). The slope is neither horizontal or vertical. It slopes diagonally downward from left to write since the slope is negative. In slope-interdept form, it is written as \(y=-\frac{1}{2} x\). In standard form, it is written as \(x+2y=0\). The slope of the line for the second train is \(\frac{7}{4}\). The slope is neither vertical or horizontal. It slopes diagonally upward from left to right since the slope is positive. In slope-intercept form, it is written as \(y=\frac{7}{4}x\). In standard form, it is written as \(-7x+4y=0\).
c. The equarion of a line in slope-intercept form of the new track would be \(y=-\frac{1}{2} x+2\).
Step-by-step explanation:
a. The way to find the coordinates is either to picture a coordinate grid in your head or to have one out in front of you for a visual. The first train travels 12 miles west, or 12 units to the left (-12), and 6 miles north, or 6 units up (6). On a coordinate plane, that would be marked as (-12,6). The second train travels 20 miles east, or 20 units to the right (20), and 35 miles north, or 35 units up (35). On a coordinate plane that would be marked as (20,35). This means that the coordinates of the first train would be (-12,6), and the coordinates of there second train would be (20,35).
b. The way to find slope is to find \(\frac{delta "x"}{delta "y"}\). "Delta" means "the change in". The equation would be \(m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\). So, let's find the slope of the first train's travel path. We'll use the origin, (0,0), and the stopping point, (-12,6), as our coordinates to find the slope. \(\frac{0-6}{0-(-12)}=\frac{0-6}{0+12}=\frac{-6}{12}=\frac{-1}{2}=-\frac{1}{2}\). That makes the slope of the first train's travel path to be (-1/2). Now, let's find the slope of the second train's travel path. We'll use the origin, (0,0), and the stopping point, (20,35), as our coordinates to find the slope. \(\frac{0-35}{0-20}=\frac{-35}{-20}=\frac{35}{20}=\frac{7}{4}\). That makes the slope of the second train's travel path to be (7/4). They want to know the equation of the travel path line in slope-intercept form and in standard form, and what the line looks like. To start off, any vertical line would be undefined, and that would be if the slope was a fraction with a denominator of zero, which doesn't occur for either train path. A horizontal line would be if the slope was zero itself and the trains were not moving, but the trains are moving, so that's not possible either. The way to determine theway a slope is diagonally is to look at whether the slope is positive or negative. The first train's path slopes negatively, so it moves diagonally downward from left to right, or diagonally upward from right to left (both the same thing). The second train's path slopes positively, so it moves diagonally upward from left to right, or diagonally downward from right to left. Finally, the equations for this part of the question. For both trains, the origin is the y-intercept, so the y-intercept does not have to be written in the slope-intercept equation. The slope-intercept form of an equation of a line is \(y=mx+b\) where "m" is the slope and "b" is the y-intercept. The "b" value is 0, so that will not be written. The slope for the first train is (-1/2), so the slope-intercept form of the first train's travel path is y=(-1/2)x. The slope of the second train is (7/4), so the slope-intercept form for this train's path is y=(7/4)x. As for standard form, subtract the "mx" value from both sides, and set the equation equal to 0. Also, get rid of any fractions. So, for the first train, standard form would be x+2y=0. And for the second train, standard form would be -7x+4y=0.
c. Parallel means having the same slope but a different y-intercept. The second train station would be placed two miles north, or two units up, from the first train station. The first train station is at (0,0), so the second train station would be placed at (0,2). That makes 2 the y-intercept for if the first train had taken the second train station's travel route. The slope for the intitial route was (-1/2), so that will stay the same. Therefore, the equation of thr route taken if the first train had come out of the second train station, if it was already built, would be y=(-1/2x)+2.
samuel has 42 pesos in his pocket which is equal to 35% of his friend's money. how much is his friend's money?
Answers
42/x = 35/100
(42)(100) = 35(x)
4200 = 35x
120 = x
A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180ft of fencing on three sides of the yard. The quadratic equation A=−2x2+180x gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.
Answers
Answer:
a) length x = 45ftb) maximum area = 4050 ft²Step-by-step explanation:
Given the quadratic equation A=−2x2+180x that gives the area A of the yard for the length x, to maximize the area of the yard then dA/dx must be equal to zero i.e dA/dx = 0
If A=−2x²+180x
dA/dx = -4x + 180 = 0
-4x + 180 = 0
Add 4x to both sides
-4x + 180 + 4x = 0 + 4x
180 = 4x
x = 180/4
x = 45
Hence the length of the building that should border the yard to maximize the area is 45 ft
To find the maximum area, we will substitute x = 45 into the modelled equation of the area i.e A=−2x²+180x
A = -2(45)²+180(45)
A = -2(2025)+8100
A = -4050 + 8100
A = 4050 ft²
Hence the maximum area of the yard is equal to 4050 ft²
Please help with the question below (please try to answer in a maximum of 1 to 5 minutes).
Answers
we have that
Prediction=7
Actual=6
substitute
\(Percent\text{ Error=}\frac{\lvert{7-6}\rvert}{\lvert{6}\rvert}*100=16.67\%\)therefore
The answer is 17%
Find the Product of (2x-3) and ( 2x+3 )
Answers
Answer:
4x^2 - 9
Step-by-step explanation:
Start with the product of the two term groupings.
(2x-3)(2x+3)
Multiply by the distributive property.
4x^2 + 6x - 6x - 9
Combine like terms.
4x^2 - 9
When there is a problem with Solver being able to find a solution, many times it is an indication of a(n)
Answers
When Solver is unable to find a solution to a problem, it can be an indication of various issues. One possible cause is that the problem may be too complex for Solver to solve within the given constraints.
In such cases, it may be necessary to adjust the problem parameters or seek alternative solutions. Another possible cause of Solver's inability to find a solution could be due to incorrect input data. This can lead to inconsistent or contradictory constraints, making it impossible for Solver to arrive at a feasible solution. Lastly, Solver may fail to find a solution due to numerical errors or limitations in the algorithm used. These issues can arise when dealing with large datasets or highly non-linear problems. In any case, when Solver is unable to find a solution, it is important to carefully examine the problem and its parameters, and consider alternative approaches. Sometimes, a small adjustment to the input data or constraints can make all the difference in arriving at a successful solution. In such cases, Solver may struggle to find an accurate or optimal solution. Ill-conditioned problems typically involve numerical instability or poor scaling, while infeasible problems occur when the given constraints cannot be satisfied simultaneously. It is crucial to analyze and refine the problem to enable Solver to find a viable solution effectively. When there is a problem with Solver being unable to find a solution, it often indicates the presence of a(n) ill-conditioned or infeasible problem.
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I need help please!!!!
Answers
Answer:
Step-by-step explanation:
Maybe A Emma, but how can we really know what Zhan, he could have any one of those in mind.. but not really figure it out with those straws :/
can someone help please !! giving brainlist to anyone who answers
Answers
Answer:
with?
Step-by-step explanation:
If A is a 9x6 matrix, what is the largest possible dimension of the row space of A? If A is a 6x9 matrix, what is the largest possible dimension of the row space of A? Explain.
Answers
If A has rank 6, then it has 6 linearly independent rows and its row space has dimension 6. If the rank of A is less than 6, then the dimension of the row space will be less than 6.
The row space of a matrix is the span of its row vectors. Therefore, the dimension of the row space is equal to the number of linearly independent rows of the matrix.
For a 9x6 matrix A, the largest possible dimension of the row space is 6. This is because there can be at most 6 linearly independent rows in a 9x6 matrix. If there were more than 6 linearly independent rows, then the matrix would have rank greater than 6, which is impossible since the maximum rank of a 9x6 matrix is 6.
For a 6x9 matrix A, the largest possible dimension of the row space is also 6. This is because the number of linearly independent rows cannot exceed the number of columns in the matrix.
Therefore, if A has rank 6, then it has 6 linearly independent rows and its row space has dimension 6.
However, if the rank of A is less than 6, then the dimension of the row space will be less than 6.
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An angle measures 22° less than the measure of its supplementary angle. What is the
measure of each angle?
Answers
Answer:
Supplementary angle= two angle that sum upto 180 degrees.
Step-by-step explanation:
\(180 - 22 = 158\)
Supplementary angle of 22° is 158°
Solve x2 + 2 = 6 by graphing the related function.
A) 2
B) there are two solutions +v8
C) no real number solutions
D) +2
Answers
The solutions of the given equation are 2 or -2
What is an equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given is an equation, x²+2 = 6
x²+2 = 6
Solving we get,
x²+2 = 6
x² = 6-2
x² = 4
x = ±2
Hence, there are two solutions of the equation 2 or -2
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Moment-Generating Function revisited [8 points] The 12 students in the home economics class of Teacher Mira wear protective gloves and each of them have different (unique) designs. Teacher Mira asked all of them to place their gloves in a box, mixed them up, and she randomly selected 6 pieces of gloves. Find the probability mass function of the random variable X, the number of matching pairs of gloves. Determine Mx (t), the moment-generating function of X and use it to find the mean and variance of X.
Answers
The probability mass function (PMF) of the random variable X, representing the number of matching pairs of gloves, needs to be determined for the given scenario.
Let's analyze the situation step by step. Since there are 12 gloves in total, the sample space consists of all possible combinations of selecting 6 gloves from the 12. The number of matching pairs can range from 0 to 6.
To calculate the PMF, we need to determine the probability of each possible outcome. The number of ways to select k matching pairs from the 6 gloves is denoted as C(k, 6), which represents a combination of k elements from a set of 6. The probability of getting k matching pairs is then equal to C(k, 6) divided by the total number of possible combinations, C(6, 12).
To find the MGF, Mx(t), we need to calculate the sum of e^(tk) multiplied by the probability of getting k matching pairs, for all values of k from 0 to 6. This will provide the MGF as a function of t.
The mean of X can be obtained by differentiating the MGF with respect to t and evaluating it at t=0. The second derivative of the MGF at t=0 gives us the variance of X.
By following these steps, the PMF, Mx(t), mean, and variance of X can be calculated for the given scenario.
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this it's just where someone who comes across this and wants points
Answers
Answer:
hows your day going? :)
The graph represents the function f(x). Which set of points lie on its inverse f^-1(x)?
Answers
Answer:
Third choice: (1, -3); (2, 2); (3, 7)
Step-by-step explanation:
There are two ways to solve this
The easiest way to solve this is to understand that the inverse of a function f(x), denoted by f⁻¹(x) is nothing but a reflection of the function about the line y = x
What that means is that if (x, y) is a specific point on the line, then the inverse point will be (y, x). Basically we are swapping the x, y values
The 3 points on f(x) and the corresponding points on f⁻¹(x) are given below:
f(x) f⁻¹(x)
------ --------
(-3, 1) (1, -3)
(2, 2) (2, 2)
(7, 3) (3, 7)
From the options given we see that the 3rd choice gives the correct values
So answer:
Third choice: (1, -3); (2, 2); (3, 7)
I have not taken the trouble of finding the equation for the line and finding the inverse of that equation since it is not needed in this question
in a race there are 7 runners. trophies for the race are awarded to the runners finishing in places 1 through 4. in how many ways can places 1 through 4 be determined?
Answers
In 5040 different ways, 7 runners can places 1 through 4 be determined if in a race there are 7 runners. trophies for the race are awarded to the runners finishing in places 1 through 4.
Define permutation.The number of possible arrangements for a given set is calculated mathematically, and this process is known as permutation. Simply said, a permutation is a term that refers to the variety of possible arrangements or orders. The arrangement's order is important when using permutations. Combinations exist when the order doesn't matter, but permutations exist when it does. A permutation could be described as an ordered combination. The following formula determines how many permutations of n objects, taken r at a time: P(n,r)=n!
Given,
Ways 7 runners can places 1 through 4 be determined:
By using permutation:
7!
7 ×6 × 5 ×4×3×2×1
5040
In 5040 different ways, 7 runners can places 1 through 4 be determined if in a race there are 7 runners. trophies for the race are awarded to the runners finishing in places 1 through 4.
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GIVING BRAINLIEST TO THE PERSON WHO ANSWERS CORRECT
Answers
The areas of the bars distort the comparison.
What is a misleading graph?A misleading graph is a graphical depiction of data that has been altered or altered in a way that is erroneous or misleading. Unintentionally, as a result of confusion or a lack of information about how to show the data correctly, a misleading graph might be produced with the purpose to mislead.
Misleading graphs could change the axes' scale, use several scales for various regions of the graph, use incorrect labelling or units, or omit or exaggerate specific data points.
Hence, the areas of the bars distort the comparison.
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