(a) (i) Φ((x + iy) + (u + iv)) = Φ((x + u) + (y + v)i) = y + v
Φ(x + iy) + Φ(u + iv) = y + v
Since these two expressions are equal, Φ is a homomorphism.
(ii) Φ(x + iy) = 0
y = 0
(iii) The image of Φ is the set of all real numbers, we have C/R ~= R.
(b) The centralizer of S in G is the set {1, r^2}.
This means that the kernel of Φ is the set of all complex numbers with an imaginary part of 0, which is the set of all real numbers. That is, ker Φ = R.
The fundamental theorem of Group Hormomorphisms states that for any group homomorphism Φ: G -> H, the kernel of Φ, ker Φ, is a normal subgroup of G, and the image of Φ, im Φ, is a subgroup of H. Furthermore, G/ker Φ is isomorphic to im Φ.
(a) (i) To show that Φ is a homomorphism, we need to show that it preserves the group operation. That is, for any two elements x + iy and u + iv in C, we need to show that Φ((x + iy) + (u + iv)) = Φ(x + iy) + Φ(u + iv). Using the definition of Φ, we have:
Φ((x + iy) + (u + iv)) = Φ((x + u) + (y + v)i) = y + v
Φ(x + iy) + Φ(u + iv) = y + v
Since these two expressions are equal, Φ is a homomorphism.
(ii) The kernel of Φ, ker Φ, is the set of all elements in C that are mapped to the identity element in R, which is 0. That is, ker Φ = {x + iy : Φ(x + iy) = 0}. Using the definition of Φ, we have:
Φ(x + iy) = 0
y = 0
This means that the kernel of Φ is the set of all complex numbers with an imaginary part of 0, which is the set of all real numbers. That is, ker Φ = R.
(iii) To prove that C/ker Φ is isomorphic to R, we can use the fundamental theorem of Group Hormomorphisms. Since ker Φ = R, we have C/R ~= im Φ. Since the image of Φ is the set of all real numbers, we have C/R ~= R.
(b) The centralizer of a subset S of a group G, denoted by C_G(S), is the set of all elements in G that commute with every element in S. That is, C_G(S) = {g : g in G, gs = sg for all s in S}. For the given groups G = D6 and S = {1,r^2}, we have:
C_G(S) = {g : g in G, g1 = 1g and gr^2 = r^2g}
= {g : g in G, g = g and gr^2 = r^2g}
= {1, r^2}
Thus, the centralizer of S in G is the set {1, r^2}.
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Gary earns $9.12 per hour at his job. He worked 24 hours last week. Calculate Gary's pay before taxes.
Gary's pay before taxes is $218.88.
Gary's pay before taxes can be calculated by multiplying his hourly rate by the number of hours he worked.
Taxes are compulsory payments made by a government organisation, whether local, regional, or federal, to people or businesses. Tax revenues are used to fund a variety of government initiatives, such as Social Security and Medicare as well as public infrastructure and services like roads and schools.
Step 1: Identify the hourly rate and number of hours worked.
Hourly rate: $9.12
Hours worked: 24
Step 2: Multiply the hourly rate by the number of hours worked.
$9.12 * 24 = $218.88
Step 3: The result is Gary's pay before taxes.
Gary's pay before taxes is $218.88.
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If \( \sin \alpha=12 / 13 \), and \( \cos \alpha=5 / 13 \), then \( \tan \alpha=? \) a) \( 5 / 12 \) b) \( 7 / 13 \) c) \( 12 / 5 \) d) \( 13 / 12 \)
The correct answer is c) \( 12 / 5 \).
We can use the relationship between the sine, cosine, and tangent of an angle to find the value of the tangent. The formula is:
\( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \)
Plugging in the given values for the sine and cosine of alpha, we get:
\( \tan \alpha = \frac{12 / 13}{5 / 13} \)
Simplifying the fraction, we get:
\( \tan \alpha = \frac{12}{5} \)
Therefore, the correct answer is c) \( 12 / 5 \).
In conclusion, if \( \sin \alpha=12 / 13 \), and \( \cos \alpha=5 / 13 \), then \( \tan \alpha=12 / 5 \).
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Suppose a projectile is launched from a tower into the air with an initial velocity of 48 feet per second. Its height, h, in feet, above the ground is modeled by the function
where t is the time, in second, since the projectile was launched and
is the initial velocity.
What tower platform height was the projectile launched from
By answering the above question, we may state that We must thus locate function the value of h that makes the equation true in order to estimate the height of the tower platform.
what is function?Mathematicians study numbers, as well as their variations, equations, related forms, and conceivable arrangements of these. Function refers to the relationship between a set of inputs, each of which has an associated output. A function is an association of inputs and outputs, where each input leads to a single, identifiable output. Each function is given a domain, codomain, or scope. The letter f is frequently used to represent functions (x). A cross is typed. The four primary types of accessible functions are on functions, one-to-one capabilities, so numerous capabilities, in capabilities, and on functions.
The projectile's starting height above the earth must be determined. Setting the time to zero (t = 0) and finding h will allow us to do this.
The formula for the projectile's height at time t is:
[tex]vt + h + -16t2 = h(t)[/tex]
where h represents the projectile's initial height and v its initial velocity.
t = 0 results in:
[tex]h(0) = -16(0)^2 + v(0) + h h(0) = h[/tex]
Hence, h, the height of the tower platform, is the projectile's beginning height.
We must thus locate the value of h that makes the equation true in order to estimate the height of the tower platform.
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Triangle XYZ has vertices X(−2, −1), Y(0, 2), and Z(2, −1)
The image after the rotation would be X(2, 1) Y(0,-2) and Z(-2, 1) according to the graph below.
What is a Graph?An ordered graphical depiction of numbers or variables is how this is described.
180 degrees in each direction results in (x,y) becoming (-x,-y). The coordinates are thus inverted to either positive or negative values depending on their respective signs.
A graph is a mathematical structure made up of a set of lines linking some pairs of VERTICES that may or may not be empty and a collection of points called VERTICES. Graphs are a common method for displaying the relationships between data. Although taking up less space, a graph accomplishes the goal of presenting data that are either too many or complex to be adequately described in the text.
There are eight different types: linear, power, quadratic, polynomial, rational, exponential, logarithmic, and sinusoidal.
Hence, it means X(−2, −1), Y(0, 2), and Z(2, −1) = X(2, 1) Y(0,-2) and Z(-2, 1).
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Complete question -
The seismic waves of a magnitude 6 earthquake are 10^2 times as great as a magnitude 4 earthquake. The wave of a magnitude 6 are 10 times as Great as a magnitude 3 earthquake
The recent earthquake in turkey was measured to be a magnitude 8. How much stronger was this earthquake compared to a magnitude 5
Earthquake in Turkey was 1000 times much stronger, compared to a magnitude 5 earthquake
What is Richter scale?The Richter scale (also known as the Richter magnitude scale) assigns magnitude numbers to quantify the energy released by an earthquake. Developed by Charles Richter in the 1930s, the Richter scale is a base 10 logarithmic scale that defines magnitude as the logarithm of the ratio of seismic wave amplitudes to smaller amplitudes.
As measured by seismographs, an earthquake measuring 5.0 on the Richter scale has 10 times more shaking amplitude than a 4.0 earthquake.
Given,
Magnitude 6 is 10² = 100 times greater than magnitude 4.
Magnitude 4 is 10 times greater than magnitude 4.
By the above data, 1 digit greater magnitude means 10 times powerful.
Earthquake on turkey is of 8 magnitude
∵ Difference of 8 magnitude and 5 magnitude = 8 - 5 = 3
∴ 8 magnitude earthquake is 1000 times greater.
Hence, 1000 times much stronger was earthquake in Turkey, compared to a magnitude 5 earthquake.
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cups pineapple juice 24 cups orange juice 46 cups seltzer 72 How many servings can be made from 108 cups of punch?
Answer:
First, find the total number of cups of liquid in the punch:
24 cups pineapple juice + 46 cups orange juice + 72 cups seltzer = 142 cups
Then, divide the total cups of liquid by the amount of liquid in each serving:
108 cups / 142 cups per serving = 0.76 servings
Therefore, approximately 0.76 servings can be made from 108 cups of punch.
The area of this trapezium is 80cm2.
Calculate the height ‘h’
The perpendicular distance between the two parallel sides, height h of the trapezium is 8cm
How to determine the height of the trapeziumIt is important to note the properties of a trapezium;
It is a 2-dimensional shapeThe bases of a trapezium are parallel The length of the diagonals is equalThe diagonals of a trapezium intersect each otherThe adjacent interior angles sum up to 180 degrees, that is ,are supplementaryThe sum of all the interior angles in a trapezium is always 360° degreesthe formula for area of trapezium is;
Area = a + b/ 2 ·(h)
Substitute the value
80 = (4 + 16)/ 2 (h)
Add the values
80 = 10h
Divide by the coefficient of h
h = 8cm
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Complete question:
The area of a trapezium is 80cm^2
The lengths of its parallel sides are 4cm and 16cm respectively. Find the perpendicular distance between the two parallel sides.
URGENT: Can you find the perimeter of each polygon and type the correct code? Please remember to type in ALL CAPS with no spaces.
PLEASEEEE HELPPP NOW!!!
Perimeter of trapezoid is 35x+2, Perimeter of square is 16x-32, Perimeter of Pentagon is 10x-20 and Perimeter of rhombus is 28x+8.
What is Polygon?A polygon is a plane figure made up of line segments connected to form a closed polygonal chain.
Perimeter of trapezoid=9x+4+7x+8+12x-2+7x-8
Add the like terms
35x+2
Perimeter of trapezoid is 35x+2
Perimeter of square= 4(4x-8)
=16x-32
Perimeter of Pentagon=5(2x-4)
=10x-20
Perimeter of rhombus=4(7x+2)
=28x+8
Hence, Perimeter of trapezoid is 35x+2, Perimeter of square is 16x-32, Perimeter of Pentagon is 10x-20 and Perimeter of rhombus is 28x+8.
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Kuro has a stack of 15 coins on a table with a volume of 8 cubic centimeters. He has stacked them perfectly straight creating a cylindrical shape. He accidentally bumps the table, causing the coins to shift to a leaning position; but are still stacked. What would be the best estimate of the volume of the stack once it has been bumped/shifted?
The best estimate of the volume of the stack once it has been bumped/shifted is 4πt³ cubic centimeters.
Describe Volume of cylinder?A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected by a curved surface. The volume of a cylinder is the amount of space occupied by the shape and is given by the formula:
Volume = πr²h
where π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.
Assuming that the coins have the same dimensions and are perfectly circular, the original stack of 15 coins formed a cylinder with a height of 15 times the thickness of a single coin, and a radius equal to the radius of a single coin.
The volume of this cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height.
Since the volume of the original stack is 8 cubic centimeters, we can set up the equation:
8 = πr²(15t)
where t is the thickness of a single coin.
Solving for r, we get:
r = √(8/15πt)
When the stack is bumped and shifts to a leaning position, the new shape will still be a cylinder, but the height will be shorter than before. Let's say that the new height is h2. We can calculate the new volume using the same formula:
V2 = πr²h2
To estimate the new height, we can use the fact that the coins are now leaning against each other, so the new height will be less than 15t. Let's say that the new height is approximately 12t.
Substituting in the values, we get:
V2 = π(√(8/15πt))²(12t)
Simplifying, we get:
V2 = 4πt³
Therefore, the best estimate of the volume of the stack once it has been bumped/shifted is 4π³ cubic centimeters.
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tiplication an(d)/(o)r division on the rational expressions ano (x^(2)-3x-18)/(x^(2)+10x+21)-:(x^(2)+3x-54)/(x^(2)-x-30)*(x^(2)+16x+63)/(x^(2)+14x+45)
The simplified rational expression is (x³-6x²+5x²)-30x+5x²-30x+25x-150)/(x³+17x²+75x+175)
To simplify the rational expression, we will need to use multiplication and division of the rational expressions. We will also need to factor the expressions in order to simplify them.
First, let's factor the expressions:
(x²-3x-18)/(x²+10x+21)-:(x²+3x-54)/(x²-x-30)*(x²+16x+63)/(x²+14x+45)
= ((x-6)(x+3))/((x+7)(x+3))-:((x+9)(x-6))/((x-6)(x+5))*((x+9)(x+7))/((x+9)(x+5))
Next, let's simplify the expressions by canceling out the common factors:
= (x-6)/(x+7)-: (x+9)/(x+5)*(x+7)/(x+5)
= (x-6)/(x+7)-: (x+9)(x+7)/(x+5)(x+5)
Now, let's multiply the expressions:
= (x-6)(x+5)(x+5)/(x+7)(x+5)(x+5) - (x+9)(x+7)(x+7)/(x+7)(x+5)(x+5)
Finally, let's subtract the expressions:
= ((x-6)(x+5)(x+5) - (x+9)(x+7)(x+7))/((x+7)(x+5)(x+5))
= (x³-6x²+5x²-30x+5x²-30x+25x-150)/(x²+17x²+75x+175)
Therefore, the simplified rational expression is:
= (x³-6x²+5x²-30x+5x²-30x+25x-150)/(x³+17x²+75x+175)
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You select a star without looking and then put it back. If you do this 16 times, what is the prediction for the number of times you will pick a star that is not yellow?
The predictiοn fοr the number οf times yοu will pick a star that is nοt yellοw is 0.999.
What is the prοbability?Prοbability is the study οf the chances οf οccurrence οf a result, which are οbtained by the ratiο between favοrable cases and pοssible cases.
Assuming that the stars cοme in οnly three cοlοrs (yellοw, red, and blue), and each cοlοr has an equal chance οf being selected, then the prοbability οf picking a star that is nοt yellοw is 2/3, and the prοbability οf picking a yellοw star is 1/3.
If we dο this experiment 16 times, we can use the binοmial prοbability fοrmula tο find the prοbability οf picking a nοn-yellοw star a certain number οf times:
[tex]\mathrm{P(x) = (n \ choose \ x) \times ^x \times (1-p)^{(n-x)}}[/tex]
where n is the number οf trials, x is the number οf times we want tο pick a nοn-yellοw star, p is the prοbability οf picking a nοn-yellοw star (2/3), and (1-p) is the prοbability οf picking a yellοw star (1/3).
Using this fοrmula, we can calculate the prοbability οf picking a nοn-yellοw star exactly 1 time, exactly 2 times, and sο οn, up tο 16 times. We can then add up these prοbabilities tο get the οverall prοbability οf picking a nοn-yellοw star at least οnce in 16 trials.
Alternatively, we can use the cοmplement rule tο find the prοbability οf picking a yellοw star in all 16 trials, and subtract this frοm 1 tο get the prοbability οf picking a nοn-yellοw star at least οnce:
P(nοn-yellοw at least οnce) = 1 - P(yellοw in all 16 trials)
= 1 - (1/3)¹⁶
≈ 0.999
Hence, the predictiοn fοr the number οf times yοu will pick a star that is nοt yellοw is 0.999.
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PLEASE HELP
Write the equation of the line with a y-intercept of (4,0) and an x-intercept of (0,6)
Answer:
y = -3/2x + 6
Step-by-step explanation:
Slope m = (y2 - y1)/(x2 - x1)
m = (6 - 0)/(0 - 4) = 6/-4 = -3/2
Slope-intercept form is y = mx + b
Substitue m = -3/2, y = 0, x = 4
0 = -3/2(4) + b
b = 6
y = -3/2x + 6
List all the possible rational roots of x^4 - 2x^3 -6x^2 + 22x - 15 = 0
The possible rational roots are:
±1/1, ±3/1, ±5/1, ±15/1, ±1/1, ±1/2, ±3/2, ±5/2, ±15/2
What are Functions?A function is a mathematical rule that assigns a unique output value for each input value. It is a set of ordered pairs where the first element is the input and the second element is the output.
To list all the possible rational roots of x⁴ - 2x³ -6x² + 22x - 15 = 0, we need to find all the factors of the constant term -15 (in this case, they are 1, -1, 3, -3, 5, -5, 15, and -15) and all the factors of the leading coefficient 1 (in this case, they are 1 and -1). Then, we form all possible fractions of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. Therefore, the possible rational roots are:
±1/1, ±3/1, ±5/1, ±15/1, ±1/1, ±1/2, ±3/2, ±5/2, ±15/2
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Select the correct answer. A company sells bikes that are light and fast. The revenue earned over a period of x years can be represented by the function, . The cost of manufacturing the bikes can be represented by the function, . Which function, , describes the profit earned from the sales of the bikes over a period of x years? A. B. C. D.
The function that describes the profit earned from the sales of the bikes over a period of x years is: [tex]P(x) = 100x^3 + 6x^2 + 9x[/tex]. The Option A is correct.
Which function describes the profit earned from the sales of the bikes?The profit earned from the sales of the bikes can be calculated by subtracting the cost of manufacturing from the revenue earned. Therefore, the function that describes the profit earned can be represented as:
[tex]P(x) = S(x) - C(x)[/tex]
Substituting the given functions for S(x) and C(x), we get:
[tex]P(x) = (100x^3 + 6x^2 + 97x + 215) - (88x + 215)[/tex]
Simplifying this expression, we get:
[tex]P(x) = 100x^3 + 6x^2 + 9x.[/tex]
Full question "A company sells bikes that are light and fast. The revenue earned over a period of x years can be represented by the function, S(x) = 100x3 + 6x2 + 97x + 215. The cost of manufacturing the bikes can be represented by the function, C(x) = 88x + 215. Which function, P(x), describes the profit earned from the sales of the bikes over a period of x years?"
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What function does this graph represent?
What is the common ratio of this geometric sequence
Answer:
what about you read well and stop cheating
Please help meeeeeee
Answer:
She earns $2,560
Step-by-step explanation:
She earns 2,560 because this is 8% of 32000. How do we know?
Multiply 32000 by 0.088, which is 8% of 100.
Bank of America charges $22 for each overdraft check. Kelsie has $370 in her account. She recently wrote checks for $46, $82, $264, and $127.
How much does she owe the bank including any overdraft charges?
Answer:
Step-by-step explanation:
Kelsie has written checks for a total of $46 + $82 + $264 + $127 = $519.
Since she only has $370 in her account, this means she has overdraft by $519 - $370 = $149.
Therefore, the bank will charge her $22 for each overdraft, which gives a total of $22 * 7 = $154.
Thus, Kelsie owes the bank a total of $519 + $154 = $673. Answer: \boxed{673}.
Omar ordered a set of purple and red pins. He received 60 pins in all. 33 of the pins were purple. What percentage of the pins were purple?
55% of the pins were purple if Omar ordered a set of purple and red pins. He received 60 pins in all. 33 of the pins were purple.
What is Algebraic expression ?
Algebraic expression can be defined as combination of variables and constants. An algebraic expression is a mathematical phrase that can contain numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.
To find the percentage of purple pins, we need to divide the number of purple pins by the total number of pins and multiply by 100.
Number of purple pins = 33
Total number of pins = 60
Percentage of purple pins = (Number of purple pins / Total number of pins) x 100
= (33 / 60) x 100
= 55%
Therefore, 55% of the pins were purple.
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pls help me with this math assignment ! give you a like
pic down below ⬇️
The three points that are not collinear are J, K and L
The three points that are not collinearIn geometry, points that are not collinear are points that do not lie on the same straight line. i.e. points that are not collinear are points that are not on the same line
Using the above as a guide, the non-collinear points are points J, K and L
How to determine the opposite raysOpposite rays are two rays that have the same endpoint and extend indefinitely in opposite directions.
Using the above definition, we have
Opposite rays = HG and HI
Another name for line JHThe line JH can be named based on its endpoints
So, another name for JH is HJ
Planes that contain HLThree points are required to form a plane
This means that the planes that contain HL can be named as GHL and JHL
There are other planes that contained HL too
Sketch of planes that intersect in line pSee attachment for the sketch
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You put $113 into a savings account that has an interest rate of 6% compounded every week.
a. How much money is in your account after 4 years?
How much interest did you earn in those 4 years?
b.
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$113\\ r=rate\to 6\%\to \frac{6}{100}\dotfill &0.06\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{weekly, thus 52} \end{array}\dotfill &52\\ t=years\dotfill &4 \end{cases}[/tex]
[tex]A = 113\left(1+\frac{0.06}{52}\right)^{52\cdot 4} \implies \boxed{A \approx 143.63}\hspace{5em}\underset{ earned~interest }{\stackrel{ 143.63~~ - ~~113 }{\boxed{\approx 30.63}}}[/tex]
Segment in a circle: XY=
There are five cities in a country, and two are connected with one road that goes straight between them. How many roads would be in this country if there were 6 cities? 7 cities? And N cities? PLEASE SOLVE 20 POINTS!
For 6 cities, the number of roads would be 9. This is because each city must be connected to every other city, and there are 5 pairs of cities that need to be connected, which means 5 roads.
What is road?Roads are a form of transportation infrastructure consisting of a prepared surface for vehicles to travel on. They are typically made from asphalt or concrete and are designed to provide a safe and efficient means of travel.
Then, the remaining 4 cities need to be connected, which means 4 more roads for a total of 9.
For 7 cities, the number of roads would be 12. This is because each city must be connected to every other city, and there are 6 pairs of cities that need to be connected, which means 6 roads. Then, the remaining 5 cities need to be connected, which means 5 more roads for a total of 12.
For N cities, the number of roads would be equal to N(N-1)/2. This is because each city must be connected to every other city, and there are N(N-1)/2 pairs of cities that need to be connected. For example, if N = 10, then 10 × 9 = 90 pairs of cities need to be connected, which means 90 roads.
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Question 1 Not yet answered Marked out of 5.00 P Flag question If the system of a linear equation has the unique solution, then the system is inconsistent Select one: True O False Question 2 Not yet
The statement "If the system of a linear equation has the unique solution, then the system is inconsistent." is False since a single solution is an indication that the equations intersect at a specific point, making it consistent.
A system of linear equations is a set of two or more linear equations that contain two or more variables. A solution to the system of linear equations is a set of values for the variables that satisfies all the equations in the system.
The system of linear equations having a single solution is a consistent system since a single solution is an indication that the equations intersect at a specific point. This contradicts the notion that the system is inconsistent. An inconsistent system is one that has no solutions.
Thus, if a system of linear equations has a unique solution, it must be consistent.
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Given that Sine theta = StartFraction 21 Over 29 EndFraction, what is the value of cosine theta, for 0 degrees less-than theta less-than 90 degrees?
Negative StartRoot StartFraction 20 Over 29 EndFraction EndRoot
Negative StartFraction 20 Over 29 EndFraction
StartFraction 20 Over 29 EndFraction
StartRoot StartFraction 20 Over 29 EndFraction EndRoot
The value [tex]Cos\theta[/tex] is 20/29 and it can be determined by using the Pythagoras theorem.
Given that,
The value [tex]Sin\theta[/tex] is [tex]\frac{21}{29}[/tex],
Where [tex]0^o < \theta < 90^o[/tex].We have to determine,
The value of [tex]Cos\theta[/tex]?
According to the question,
To determine the value of [tex]Cos\theta[/tex] following all the given below.
The formula of [tex]Sin\theta[/tex] is,
[tex]Sin\theta=\dfrac{Perpendicular}{Hypotenuse}[/tex]
On comparing to the given value of [tex]Sin\theta[/tex] is,
[tex]Sin\theta=\dfrac{Perpendicular}{Hypotenuse}[/tex]
[tex]sin\theta=\dfrac{21}{29}[/tex]
Here, Hypotenuse = 29, and Perpendicular = 21
Then,
The formula of [tex]Cos\theta[/tex] is,
[tex]Cos\theta=\dfrac{Base}{Hypotenuse}[/tex]
Let, the base be x,
The value of base finding by using Pythagoras theorem,
[tex](Hypotenuse)^2=(Perpendicular)^2+(Base)^2[/tex]
[tex](29)^2=(21)^2+(x)^2[/tex]
[tex]841=441+x^2[/tex]
[tex]x^2=841-441[/tex]
[tex]x^2=400[/tex]
[tex]x=20[/tex]
The value of the base is 20.
Therefore,
[tex]Cos\theta=\dfrac{Base}{Hypotenuse}[/tex]
[tex]cos\theta=\dfrac{20}{29}[/tex]
Hence, The value [tex]Cos\theta[/tex] is 20/29.
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Find the length of each leg,hyp,opp,adj
The length of the legs are 2 units and 3.464 units
How to determine the length of the legsFrom the question, we have the following parameters that can be used in our computation:
The triangle
The legs can be calculated using sine and cosine ratios
Using the above as a guide, we have the following:
sin(30) = PQ/Hyp
cos(30) = RQ/Hyp
Substitute the known values in the above equation, so, we have the following representation
PQ/4 = 0.5
RQ/4 = 0.8660
So, we have
PQ = 4 * 0.5
RQ = 4 * 0.8660
Evaluate
PQ = 2
RQ = 3.464
Hence, one of the leg is 3.464 units
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20. Find a \( 2 \times 2 \) matrix \( A \) for which \[ \left[\begin{array}{ll} 2 & 5 \\ 1 & 3 \end{array}\right] A=\left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right] \text {. } \]
The matrix \( A \) that satisfies the given equation is \[ A=\left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right]. \]
To find a \( 2 \times 2 \) matrix \( A \) such that \[ \left[\begin{array}{ll} 2 & 5 \\ 1 & 3 \end{array}\right] A=\left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right] \text {. } \], we can solve for A by multiplying both sides of the equation by the inverse of the left matrix. The inverse of \[ \left[\begin{array}{ll} 2 & 5 \\ 1 & 3 \end{array}\right] \] is \[ \left[\begin{array}{ll} -3 & 2 \\ 5 & -2 \end{array}\right]. \] Multiplying both sides of the equation by this inverse gives \[ \left[\begin{array}{ll} -3 & 2 \\ 5 & -2 \end{array}\right] \left[\begin{array}{ll} 2 & 5 \\ 1 & 3 \end{array}\right]A= \left[\begin{array}{ll} -3 & 2 \\ 5 & -2 \end{array}\right] \left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right], \] which simplifies to \[ A=\left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right]. \] Thus, the matrix \( A \) that satisfies the given equation is \[ A=\left[\begin{array}{rr} -1 & 0 \\ 4 & 2 \end{array}\right]. \]
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PLSS HELP IT DUE TONIGT
Answer:
mean 36.2 median is 33
Step-by-step explanation:
For median you put the numbers in least to greatest and then find the middle which in this case is 33 and 34 so 33 is in between those two numbers so that would be the median. For the mean you add all the number together which is 362 and then divide by the number of numbers that there is which is 10
Find the scalar equation for the plane passing through the
points P1=(−3, 5, 3), P2=(−5, 0, 6), and P3=(−1, 10, −1).
The scalar equation of the plane is -5x - 2y = 5.
To find the scalar equation for the plane passing through the points P1=(−3, 5, 3), P2=(−5, 0, 6), and P3=(−1, 10, −1), we need to follow these steps:
1. Find the vectors P1P2 and P1P3 by subtracting the corresponding coordinates of the points:
P1P2 = P2 - P1 = (-5 - (-3), 0 - 5, 6 - 3) = (-2, -5, 3)
P1P3 = P3 - P1 = (-1 - (-3), 10 - 5, -1 - 3) = (2, 5, -4)
2. Find the normal vector to the plane by taking the cross product of P1P2 and P1P3:
n = P1P2 x P1P3 = (-5 * (-4) - 3 * 5, 3 * 2 - (-2) * (-4), -2 * 5 - (-5) * 2) = (-5, -2, 0)
3. Find the scalar equation of the plane by substituting one of the points and the normal vector into the general equation of a plane:
ax + by + cz = d
-5 * (-3) + (-2) * 5 + 0 * 3 = d
d = 5
Therefore, the scalar equation of the plane is -5x - 2y = 5.
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Are (9g – 11 + 10g) - (12 - 11g+ 13) and -3(-10g + 12) equivalent expressions? Explain.
Yes, these expressions are equivalent expressions.
What are expressions?A sentence qualifies as a mathematical expression if it comprises one or more mathematical operations, at least two numbers, or variables. Let's examine the writing style of expressions. A number is 6 times larger than the other number, x, by a factor of 2. The mathematical phrase x/2 + 6 represents this claim. Complex riddles are solved using mathematical expressions.
In the given question,
Expressions are as follows:
(9g-11+10g) - (12-11g+13)
Simplifying it, we get:
30g -36
In the 2nd expression given,
-3(-10g+12) and after simplifying:
30g-36
As the expressions after simplifying are equal to each other, we can say that these are equivalent expressions.
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