Answer:
Yes, both students A and B verified the identity properly.
Because in the final step, the expressions on both sides of the equation were equal ( RHS = LHS) and the correct trig identities were utilized in the course of the solution.
Explanation:
Given the below identity;
\(\mleft(\cot x\mright)\mleft(\cos x\mright)=cscx-\sin x\)Let's analyze the steps Student A used in proving the above identity;
Step 1:
\(\frac{\cos x}{\sin x}\cdot\cos x=\csc x-\sin x\)Recall;
\(\begin{gathered} \tan x=\frac{\sin x}{\cos x}\text{ and cot x }=\frac{1}{\tan x} \\ \therefore\text{cot x }=\frac{1}{\frac{\sin x}{\cos x}}=1\div\frac{\sin x}{\cos x}=1\cdot\frac{\cos x}{\sin x}=\frac{\cos x}{\sin x} \end{gathered}\)Step 2:
\(\frac{\cos^2x}{\sin x}=\csc x-\sin x\)Step 3:
\(\frac{1-\sin ^2x}{\sin x}=\csc x-\sin x\)The below trig identity was applied here;
\(\begin{gathered} \cos ^2x+\sin ^2x=1 \\ \therefore\cos ^2x=1-\sin ^2x \end{gathered}\)Step 4:
\(\frac{1}{\sin x}-\frac{\sin ^2x}{\sin x}=\csc x-\sin x\)Step 5:
\(\csc x-\sin x=\csc x-\sin x\)Note that;
\(\begin{gathered} \frac{1}{\sin x}=\csc x \\ \text{and} \\ \frac{\sin^2x}{\sin x}=\sin x \end{gathered}\)We can see from the above that Student A verified the identity properly because in the final step, the expressions on both sides of the equation were equal ( RHS = LHS) and the correct trig identities were utilized.
Let's analyze the steps of Student V;
Step 1:
Recall that;
\(\csc x=\frac{1}{\sin x}\)So Step 1 is given as;
\(\cot x\cos x=\frac{1}{\sin x}-\sin x\)Step 2:
Note that;
\(\frac{\sin^2x}{\sin x}=\sin x\)So Step 2 is given as;
\(\cot x\cos x=\frac{1}{\sin x}-\frac{\sin^2x}{\sin x}\)Step 3:
Using sin x which is the LCM to multiply through, we'll have;
\(\cot x\cos x=\frac{1-\sin^2x}{\sin x}\)Step 4:
Recall the below trig identity;
\(\begin{gathered} \cos ^2x+\sin ^2x=1 \\ \therefore\cos ^2x=1-\sin ^2x \end{gathered}\)Applying the above identity, we'll have;
\(\cot x\cos x=\frac{\cos^2x}{\sin x}\)Step 5:
Note that;
\(\cos ^2x=\cos x\cdot\cos x\)So we'll have;
\(\cot x\cos x=\frac{\cos x}{\sin x}\cdot\cos x\)Step 6;
Recall that;
\(\begin{gathered} \tan x=\frac{\sin x}{\cos x} \\ \text{and} \\ \cot x=\frac{1}{\tan x}=\frac{1}{\frac{\sin x}{\cos x}}=1\div\frac{\sin x}{\cos x}=1\cdot\frac{\cos x}{\sin x}=\frac{\cos x}{\sin x} \\ \therefore\cot x=\frac{\cos x}{\sin x} \end{gathered}\)So we'll have;
\(\cot x\cos x=\cot x\cos x\)We can see from the above that Student B verified the identity properly because in the final step, the expressions on both sides of the equation were equal ( RHS = LHS) and the correct trig identities were utilized.
PLEASE HELP! urgent! Write the equation of a line in a slope-intercept form and standard form
Answer:
\(y = \frac{3}{4}x -3\) --- Slope intercept
\(4y - 3x =- 12\) --- Standard form
Step-by-step explanation:
Given
The attached linear graph
Required
The equation
Select any two points on the line
\((x_1,y_1) = (0,-3)\)
\((x_2,y_2) = (4,0)\)
Next, calculate the slope (m)
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
So, we have:
\(m = \frac{0 --3}{4 - 0}\)
\(m = \frac{3}{4}\)
The slope intercept is then calculated using:
\(y = m(x - x_1) + y_1\)
So, we have:
\(y = \frac{3}{4}(x - 0) -3\)
\(y = \frac{3}{4}(x) -3\)
\(y = \frac{3}{4}x -3\)
Multiply through bu 4
\(4y = 3x - 12\)
Subtract 3x from both sides
\(4y - 3x =- 12\) --- Standard form
Create and expression that is equivalent to 3x+2x+x using the fewest terms
Answer:
6x
Step-by-step explanation:
add the terms to make one term
PLEASE WHOEVER DOES IT GETS BRAINLIEST
Answer:
first one on top to the left
Step-by-step explanation:
Need help on this guys, help pls.
Answer:
1,740 261
20+3
Final answer 23
Step-by-step explanation:
Sorry If I am wrong.
Find the graph of Y equals negative X squared plus 5X -6 by making a table of a few values and plotting the points
Given the function:
\(y=-x^2+5x-6\)Let's find the graph of the function by making a table of few values and plotting the points.
Let's find 3 different points by inputting random values for x and solve for y.
• When x = 0:
Substitute 0 for x and solve for y
\(\begin{gathered} y=-0^2+5(0)-6 \\ y=0+0-6 \\ y=-6 \\ \\ \text{ Here, we have the point:} \\ (0,\text{ -6\rparen} \end{gathered}\)• When x = 2:
Substitute 2 for x and solve for y
\(\begin{gathered} y=-(2)^2+5(2)-6 \\ y=-4+10-6 \\ y=0 \\ \\ Here,\text{ we have the point:} \\ (2,\text{ 0\rparen} \end{gathered}\)• When x = 3:
Substitute 3 for x and solve for y
\(\begin{gathered} y=-(3)^2+5(3)-6 \\ y=-9+15-6 \\ y=0 \\ \\ \text{ Here, we have the point:} \\ (3,\text{ 0\rparen} \end{gathered}\)• When x = 1:
\(\begin{gathered} y=-(1)^2+5(1)-6 \\ y=-1+5-6 \\ y=-2 \\ \\ We\text{ have the point:} \\ (1,\text{ -2\rparen} \end{gathered}\)Therefore, we have the table of values representing the function:
Now, let's plot the points to find the graph:
(0, -6), (1, -2), (2, 0), (3, 0)
We have the graph below:
Solve the triangle. Round to the nearest tenth.
Find c aswell
Answer:
Set your calculator to Degree mode.
a² = 17² + 20² - 2(20)(17)cos(89°)
a² = 677.13236
a = 26.0 in.
sin(89°)/26.02177 = (sin B)/17
sin B = 17sin(89°)/26.02177
B = 40.8°
C = 50.2°
a triangular prism (like a box of toblerone) of mass m, whose two ends are equilateral triangles parallel to the xy plane with side 2a, is centered on the origin with its axis along the z axis. find its moment of inertia for rotation about the z axis. without doing any integrals write down and explain its two products of inertia for rotation about the z axis.
The moment of inertia for rotation about the z axis is \(I & =\frac{a^2 M}{3}\).
A triangular prism of mass m, whose two ends are equilateral triangles.
Two ends are equal.
The xy plane with side 2a, is centered on the origin with its axis along the z axis.
Find the moment of inertia for rotation the z-axis.
It is divide into 4 triangles with side 'a' k one side is 3cm.
\($$\begin{aligned}& I_{\text {BiG }}=16 I_{\text {small }} . \\& I_{\text {BIG }}=4 I_{\text {small }}+P_{A T}\end{aligned}$$\)
It is fact that the big triangle is made up from four small triangles. connected with parallel axis theorem.
Now, we have to find the distance from the center of mas of small outer triangles to the origin. so, that it is PAT.
\($$\therefore z=2 \sqrt{x^2-\frac{a}{2} ^2}$$\)
\($$\begin{aligned}& =2 \sqrt{\left(\frac{a}{\sqrt{3}}\right)^2-\frac{a^2}{4}} \\& =2 \sqrt{\frac{a^2}{3}-\frac{a^2}{4}} \\I_{\text {Big }} & =\frac{\sqrt{3}}{3} a \cdot \\& \left.=4 I_{\text {small }}+3 m \frac{\sqrt{3}}{3} a\right)^2 \\I_{\text {Big }} & =4 I_{\text {small }}+\frac{3 m}{4} \frac{1}{3} a^2 \\& =\frac{a^2}{4}\end{aligned}$$\)
⇒ Now, by using the \($I_{\text {Big }}=I$\)
⇒\(I & =4\left(\frac{\sqrt{2}}{16}\right)+\frac{a^2 M}{4} \\\)
⇒\(4 I & =I+a^2 m\)
⇒\(4 I-I & =a^2 m \\\)
⇒\(3 I & =a^2 M . \\\)
⇒\(I \quad & =\frac{a^2 M}{3}\)
Therefore, the moment of inertia is \(I & =\frac{a^2 M}{3}\).
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If each quadrilateral below is a rectangle, find the missing measures.
Please helpppp
The measures of the angles are JKN = 27 degrees and JNK = 126
How to determine the measures of the anglesThe complete question is added as an attachment
From the question, we have the following parameters that can be used in our computation:
The rectangle
On the rectangle, we have
JKN = NJK
Using the above as a guide, we have the following:
JKN = 27 degrees
Next, we use the sum of angles in a triangle theorem
The sum of angles in a triangle is 180
Mathematically, this is represented as
J + N + K = 180
So, we have the following representation
JNK = 180 - 27 * 2
Evaluate
JNK = 126
Hence, the measure of the angle JNK is 126 degrees
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PLEASE HELP ASAP
The diagrams show a polygon and the image of the polygon after a transformation.
Use the diagrams to determine which statements are true. Select all statements that are true.
The correct statements regarding the transformations are given as follows:
Parallel lines will always be parallel after a reflection.Lines that are not parallel will never be parallel after a translation.What are transformations on the graph of a function?Examples of transformations are given as follows:
A translation is defined as lateral or vertical movements.A reflection is either over one of the axis on the graph or over a line.A rotation is over a degree measure, either clockwise or counterclockwise.For a dilation, the coordinates of the vertices of the original figure are multiplied by the scale factor, which can either enlarge or reduce the figure.Parallel lines are lines that have the same slope, that is, lines that do not intercept, and the transformations do not change whether the lines are parallel or not.
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A rectangular garden is 8 ft longer than it is wide. Its area is 425 ft2. What are its dimensions?
Answer:
17 ft.
Step-by-step explanation:
Because 17x25=425
And 25-8=17
What’s the value of x in both a & b?
Answer:
a) 7
b) 2 2/7
Step-by-step explanation:
A) 2/7=2/x
x=7
B) 2/7=x/8
cross multiply
16=7x
x=16/7, or 2 2/7, or 2.2857
what is the remainder for the synthetic division problem? (image attached)
Answer:
D
Step-by-step explanation:
3 | 1 5 - 8 6
↓ 3 24 48
----------------------
1 8 16 54 ← Remainder
determine the general solution of cos2X -7cosX -3=0
Answer:
x=2pi/3 +2pi n, 4pi/3 +2pi n for all integar of n.
Step-by-step explanation:
The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 23.5 for a sample of size 775 and standard deviation 12.2. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 95% confidence level). Enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
The 95% confidence interval for the effectiveness of the blood-pressure drug is given as follows:
\(22.6 < \mu < 24.4\)
How to obtain the confidence interval?The mean, the standard deviation and the sample size for this problem, which are the three parameters, are given as follows:
\(\overline{x} = 23.5, \sigma = 12.2, n = 775\)
Looking at the z-table, the critical value for a 95% confidence interval is given as follows:
z = 1.96.
The lower bound of the interval is then given as follows:
\(23.5 - 1.96 \times \frac{12.2}{\sqrt{775}} = 22.6\)
The upper bound of the interval is then given as follows:
\(23.5 + 1.96 \times \frac{12.2}{\sqrt{775}} = 24.4\)
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help me please im stuck
Look at Jerrys monthly budget below. What percentage of Jerrys net income is spent on transportation?
Net Monthly Income: 2,970.00
Housing: $750.00
Utilities: $120.00
Food: $450.00
Transportation: $650.00
Savings: $300.00
Other Expenses: $500.00
Total Monthly Expenses: 2,970.00
A. 28%
B. 25.3%
C. 86.7%
D. 21.9%
Answer:
D. 21.9%
Step-by-step explanation:
(650÷2970)x100= 21.88% rounded off 21.9%
joe paid 14.00 for a broard game. THis is 70% of the orginal price. What was the orginal price?
Answer:
20.00
Step-by-step explanation:
14 is 70% of X
70/100 × X = 14
X = 14 × 100/70
X = 20.00
What is the inverse of the following conditional? If Tracy is not going to Quinn's candy shop then she is not going to buy sour straws.
In order to find the inverse of this conditional, we need to negate both the first and second statement.
So we have:
First statement: "Tracy is not going to Quinn's candy shop"
Second statement: "she is not going to buy sour straws"
Negating these, we have:
First statement: "Tracy is going to Quinn's candy shop"
Second statement: "she is going to buy sour straws"
So the inverse will be:
"If Tracy is going to Quinn's candy shop then she is going to buy sour straws."
Explain a time in your life you underwent metamorphosis
Answer: Maybe when you Grew from a Baby to a toddler because that would be you growing over time
Step-by-step explanation:
I Hope This Helps!
The average diameter of a tree is measured over time. The equation y=3/5x, where x is the time in years and y is the diameter in inches, models the relationship. What is the predicted diameter, in inches, of the three after 12 years?
Please help me I will mark brainliest! The ratio of the number of boys to the number of girls in a school is 3:4. One-third of the boys and three-eighths of the girls wear spectacles, If there are 612 pupils who do not wear spectacles, a)find the total number of the pupils in the school, and b) how many more girls than boys are there in the school
Answer:
a) 952
b) 136
Step-by-step explanation:
Ratio of b:g = 3:4, based on this we have:
Number of boys = 3xNumber of girls = 4xTotal number of pupils = 3x+4x = 7xNumber of spectacle wearers:
1/3*3x + 3/8*4x = x + 3/2x = 2.5xNumber of those not wearing spectacles:
7x - 2.5x= 4.5xAnd this number equals to 612, then we can find the value of x:
4.5 x = 612x= 612/4.5x= 136a) Total number of pupils:
7x = 7*136 = 952b) The difference in the number of boys and girls:
4x-3x= x = 136Answer:
total number of students: 952
number of girls more than boys :136 more girls
Step-by-step explanation:
1/3 of boys +3/8 girls= spectacles
612 people do not wear spectacles
3:4= boys: girls
total number of students
3+4=7
boys + girls = total ratio
7= total ratio
1/3×3=1 3/8×4=3/2
1+3/2=5/2
7-5/2=9/2 9/2=612 students
If 9/2=612 Then 7=?
7= 7÷ 9/2×612
=952 people
Girls more than boys
if 7= 952
3= 3/7 × 952=408 boys
if 7 = 952
4= 4/7 ×952=544girls
Girls - boys
544- 408 = 136 girls
18) What is the slope of the line that contains points (–6, –6) and (–3, 1)?
The slope of the line is 7/9
How to determine the slope of the lineIt is important to note that the equation of a line is represented as;
y = mx + c
Where;
y is a point on the linem is the slope of the linex is a point on the x - axisc is the intercept of the y-axisThe formula for calculating the slope of a line is expressed as;
Slope, m = y₂ - y₁/x₂ - x₁
Now, let's substitute the values into the formula from the points given we have;
Slope, m =1 -(-6)/ -3 - (-6)
expand the bracket
Slope, m = 1 + 6/ 3 + 6
add the values
Slope, m = 7/9
Hence, the value is 7/9
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Kallie works at a pet store. Part of her job is to add the correct amount of water conditioner to each fish tank the list below provides information about the number of fish tanks and the amount of water conditioner she uses.
1: there are 12 fish tanks that need water conditioner
2: Each fish tank is filled with 20 quarts of water
3: for every 10 gallon of water, Kallie uses 1 teaspoon of water conditioner
What is the total number of teaspoons of water conditioner kallie will use for all the water in 12 fish tanks?
Therefore, Kallie will need to use a total of 6 teaspoons of water conditioner for all the water in 12 fish tanks.
What is equation?An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides separated by an equal sign, with the expression on the left being equal to the expression on the right. Equations are used to represent various relationships and situations in mathematics and other fields, and are solved to find the values of variables that make the equation true.
Here,
First, we need to determine the total amount of water in all 12 fish tanks. Since each fish tank is filled with 20 quarts of water, the total amount of water in all 12 fish tanks is:
12 fish tanks * 20 quarts per fish tank = 240 quarts of water
To convert quarts to gallons, we divide by 4 (since there are 4 quarts in a gallon):
240 quarts / 4 quarts per gallon = 60 gallons of water
Now, we can use the given conversion factor to determine how many teaspoons of water conditioner are needed for 60 gallons of water. Since for every 10 gallons of water, Kallie uses 1 teaspoon of water conditioner, for 60 gallons of water, she will use:
60 gallons / 10 gallons per teaspoon = 6 teaspoons of water conditioner
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Suppose an account pays 6% interest that is compounded annually. At the beginning of each year, $2,000 is deposited into the account (starting with $2,000 for the first year).
Using the future value formula, it is found that the value of the account after the tenth deposit is of $26,361.59.
What is the future value formula?The future value formula is given by:
\(V(n) = P\left[\frac{(1 + r)^n - 1}{r}\right]\)
In which:
P is the payment.n is the number of payments.r is the interest rate.For this problem, the parameters are given as follows:
P = 2000, r = 0.06, n = 10.
Hence the value of the account will be of:
\(V(10) = 2000\left[\frac{(1 + 0.06)^{10} - 1}{0.06}\right]\)
V(10) = $26,361.59.
What is the missing information?The complete problem is:
"Suppose that there is an account that pays 6% interest that is compounded annually. At the beginning of each year, $2,000 is deposited into the account (starting with $2,000 for the first year).
What will be the value of the account after the tenth deposit if no withdrawals or additional deposits are made?"
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115 degrees (5x-10degrees) what does x equal show your work
Answer: 21
Step-by-step explanation:
115-10=105
105/5=21
Match the vocabulary word with the correct definition. 1. axis of rotation a line around which a two-dimensional shape is rotated or revolved to create a three-dimensional shape 2. cross section the two-dimensional shape that results from cutting a three-dimensional shape with a plane 3. solid of revolution the three-dimensional shape that is produced when a two-dimensional shape is rotated or revolved around a line
Answer:
See step by step explanation
Step-by-step explanation:
Matching the vocabulary word with the correct definition, we get;
1. Axis of Rotation: a line around which a two-dimensional shape is rotated or revolved to create a three-dimensional shape.
It is a straight line through a rigid body when all its part are in rotation.
2. Cross section: the two-dimensional shape that results from cutting a three-dimensional shape with a plane. It is a internal view of an object that is made by cutting through it.
3. Solid of Revolution: the three-dimensional shape that is produced when a two-dimensional shape is rotated or revolved around a line. In mathematics, it is a solid shape that is obtained by revolving a curve around an axis.
Un cuerpo de 30 kg de masa se encuentra a una altura de 5 metros en relación al suelo, siendo g= 9,8 m/s2, calcula la energía potencial del cuerpo
Answer:
bro speak english
SPEAK ENGLISH
Help plssssss I really need the answer asap! I’d really appreciate it
Find the slope of each line passing through the given points using the slope formula.
(-3, -10) and (-1, -1) URGENT Please answer
Slope (m)= y\x= 9/2 = 4.5
your equation for the line is
y=4.5x+3.5
Just number 9 with work
The answer of the given question based on rational parent function , to identify a correct transformation of the function, The correct answer choice is (c) Left 5.
What is Function?A function is mathematical relationship that assigns unique output value for each input value. It describes relationship between two variables, where output value is dependent on input value. A function can be represented by equation, a graph, or table. In its simplest form, function takes input value, performs a specified operation on that value, and produces corresponding output value.
Functions are used to model real-world phenomena, to solve problems, and to analyze and understand complex systems. They are fundamental concept in mathematics and are used in wide variety of fields, like science, engineering, economics, and finance.
The given equation is y = 12/(x + 5), which represents the rational parent function. To identify a correct transformation of the function, we can consider the effect of each transformation on the parent function.
a) Vertical reflection: This transformation would change the sign of the function, resulting in y = -12/(x + 5). This is not the given equation, so it is not a correct transformation.
b) Vertical stretch by 12: This transformation would multiply the function by 12, resulting in y = 12(12/(x + 5)) = 144/(x + 5). This is not the given equation, so it is not a correct transformation.
c) Left 5: This transformation would shift the function to the left by 5 units, resulting in y = 12/(x + 10). This is a correct transformation of the function, as it represents a horizontal shift of 5 units to the left.
d) Down 5: This transformation would shift the function down by 5 units, resulting in y = 12/(x + 5) - 5. This is not the given equation, so it is not a correct transformation.
Therefore, the correct transformation of the rational parent function as described by the given equation is a horizontal shift of 5 units to the left. The answer choice is (c) Left 5.
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The correct transformation of the function on rational parent function as described by the given equation. The correct answer choice is (c) Left 5.
What is Function?A function is mathematical relationship that assigns unique output value for each input value. It describes relationship between two variables, where output value is dependent on input value. A function can be represented by equation, a graph, or table. In its simplest form, function takes input value, performs a specified operation on that value, and produces corresponding output value.
The given equation is y = 12/(x + 5), which represents the rational parent function. To identify a correct transformation of the function, we can consider the effect of each transformation on the parent function.
a) Vertical reflection: This transformation would change the sign of the function, resulting in y = -12/(x + 5). This is not the given equation, so it is not a correct transformation.
b) Vertical stretch by 12: This transformation would multiply the function by 12, resulting in y = 12(12/(x + 5)) = 144/(x + 5). This is not the given equation, so it is not a correct transformation.
c) Left 5: This transformation would shift the function to the left by 5 units, resulting in y = 12/(x + 10). This is a correct transformation of the function, as it represents a horizontal shift of 5 units to the left.
d) Down 5: This transformation would shift the function down by 5 units, resulting in y = 12/(x + 5) - 5. This is not the given equation, so it is not a correct transformation.
Therefore, the correct transformation of the rational parent function as described by the given equation is a horizontal shift of 5 units to the left. The answer choice is (c) Left 5.
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