Pls help
Prove the trigonometric identity
csc x cos x/tan x+cot x =cos^2 x
Drag the expressions into each box to complete the proof.
Answers
Answer:
Here is the proof of the trigonometric identity
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2 x\)
Given:
\(\frac{\csc x \cos x}{\tan x + \cot x}\)
To prove:
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2x\)
Step 1: Rewrite all trigonometric functions in terms of sine and cosine.
\(\frac{\csc x \cos x}{\tan x + \cot x} \\ = \frac{\frac{1}{\sin x} \cos x}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} \\ = \frac{\cos x}{\sin x \cdot \frac{\sin x}{\cos x} + \cos x \cdot \frac{\cos x}{\sin x}} \\ \\ = \frac{\cos x}{\frac{\sin^2 x + \cos^2 x}{\sin x \cos x}} \\ \\ = \frac{\cos x}{\frac{1}{\sin x \cos x}} \\ \\ = \cos x \cdot \sin x \cos x \\ \\ = \cos^2 x \)
Step 2:Simplify the expression.
\(\cos^2 x = \cos^2 x\)
Conclusion:Therefore,
\(\frac{\csc x \cos x}{\tan x + \cot x} = \cos^2 x.\)
Hence Proved:
Answer:
\(\dfrac{ \csc x \cos x}{\tan x + \cot x}=\cos^2 x\)\(\textsf{Rewrite $\csc x=\dfrac{1}{\sin x}}, \tan x=\dfrac{\sin x}{\cos x}$, and $\cot x=\dfrac{\cos x}{\sin x}$:}\)
\(\boxed{\dfrac{ \dfrac{1}{\sin x} \cdot \cos x}{\dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x }{\sin x + \cos x}+\dfrac{\cos^2 x }{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x + \cos^2 x}{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{1}{\sin x + \cos x}}}=\cos^2 x\)
\(\boxed{\dfrac{\cos x}{\sin x} \cdot \sin x \cos x}=\cos^2 x\)
\(\cos^2 x=\cos^2 x\)
Step-by-step explanation:
Given trigonometric identity:
\(\dfrac{ \csc x \cos x}{\tan x + \cot x}=\cos^2 x\)
\(\textsf{Use the identities\;\;$\csc x=\dfrac{1}{\sin x}$\;,\;$\tan x = \dfrac{\sin x}{\cos x}$\;\;and\;\;$\cot x=\dfrac{\cos x}{\sin x}$}:\)
\(\textsf{Rewrite $\csc x=\dfrac{1}{\sin x}}, \tan x=\dfrac{\sin x}{\cos x}$, and $\cot x=\dfrac{\cos x}{\sin x}$:}\)
\(\boxed{\dfrac{ \dfrac{1}{\sin x} \cdot \cos x}{\dfrac{\sin x}{\cos x} + \dfrac{\cos x}{\sin x}}}=\cos^2 x\)
Simplify the numerator and make the fractions in the denominator like fractions:
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x }{\sin x + \cos x}+\dfrac{\cos^2 x }{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Apply\;the\;fraction\;rule\;\;$\dfrac{a}{b}+\dfrac{c}{b}=\dfrac{a+c}{b}$\;to\;the\;denominator}:\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{\sin^2 x + \cos^2 x}{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Use\;the\;identity\;\;$\sin^2x+\cos^2x=1$}:\)
\(\boxed{\dfrac{\dfrac{\cos x}{\sin x}}{\dfrac{1}{\sin x + \cos x}}}=\cos^2 x\)
\(\textsf{Apply\;the\;fraction\;rule\;\;$\dfrac{a}{\frac{b}{c}}=a \cdot \dfrac{c}{b}$}:\)
\(\boxed{\dfrac{\cos x}{\sin x} \cdot \sin x \cos x}=\cos^2 x\)
Cancel the common factor sin x, and apply the exponent rule aa = a²:
\(\cos^2 x=\cos^2 x\)
Related Questions
Baby talk: In a sample of 69 children, the mean age at which they first began to combine words was 16.44 months. with a standard deviation of 9.47 months. Part 1 out of 2 Construct a confidence interval for the mean age at which children first begin to combine words. Round the answers to two decimal places. A 98% confidence interval for the mean age is:__________
Answers
Answer:
A 98% confidence interval for the mean age is:
= ( 13.78, 19.10) months
Step-by-step explanation:
Confidence interval can be defined as a range of values so defined that there is a specified probability that the value of a parameter lies within it.
The confidence interval of a statistical data can be written as.
x+/-zr/√n
Given that;
Mean x = 16.44 months
Standard deviation r = 9.47 months
Number of samples n = 69
Confidence interval = 98%
z(at 98% confidence) = 2.33
Substituting the values we have;
16.44+/-2.33(9.47/√69)
16.44+/-2.33(1.140054028722)
16.44+/-2.656325886922
16.44+/-2.66
= ( 13.78, 19.10) months
Therefore, A 98% confidence interval for the mean age is:
= ( 13.78, 19.10) months
what is the place value of 8 in fullowing number? 348000
Answers
Answer: 8000
Step-by-step explanation:
The 8 in the number 348,000 is in the thousands place.
So it represents the value of 8,000 .
Use deductive reasoning to tell whether each statement is true or false. If it is
false, give a counterexample.
22. For all real numbers a and b, a - b = -b + a.
23. For all real numbers p, q and r, p - 4-r=p-r-q.
24. For all real numbers x, y and z, (x + y) + z = z + (x + y).
25. For all real numbers m and n, m. n = n. m.
Answers
so the answer for this particular question is unknown because I havent done this course before
de su
1
5
El tanque de gasolina del auto de Chris está lleno a
3
de su capacidad. Después de comprar 11 galones de gasolina, está a
capacidad. ¿Cuántos galones caben en el tanque del auto de Chris?
Answers
Jk jk lol
PLEASE HELP FAST. Maggie pays for a 6-month subscription to a video streaming channel. Because she purchased 6 months at once, she receives a $10 discount. Her final price after the discount is $43.94. What was the original price per month (H)?
Answers
Answer:
53.94$
Step-by-step explanation:
Answer:
53.94$
Step-by-step explanation:
In the revenue equation R = –360p2 + 28,800p, what is the maximum revenue that can be expected?
Group of answer choices
Answers
The revenue equation is given by:
R = –360p^2 + 28,800p
To find the maximum revenue that can be expected, we need to find the vertex of the parabola represented by this equation. The vertex of a parabola is given by the formula:
Vertex = (-b/2a, f(-b/2a))
where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, a = -360 and b = 28,800.
The x-coordinate of the vertex is given by:
x = -b/2a = -28,800/(2*(-360)) = 40
Substituting x = 40 into the revenue equation gives:
R = –360(40)^2 + 28,800(40) = 576,000
Therefore, the maximum revenue that can be expected is $576,000.
Again let me know what other ones you need help with
What is the percent of increase from 400,000 to 500,000?
Answers
Answer:
25%
Step-by-step explanation:
500,000-400,000 = 100,000
100,000 / 400,000 = .25 or 25%
Answer: the answer is 25%
Trapezoid WXYZ is located at W(2, 1) X(3,
5) Y(6.5) and Z17. 1). After being reflected
over the x-axis, what are the new
coordinates of Point W?
Answers
W (2,-1)
when flipping over the x axis, the rule is that the x doesn’t change while the y coordinate becomes negative
i hope i have been useful buddy.
good luck ♥️♥️♥️♥️♥️.
I NEED HELP PLS I FORGOT HOW TO DO THIS HELP
Answers
Answer:
Step-by-step explanation:
how many shapes of size A are in B ?
I counted 9.5
then just multiply the count * the size of A
got it?
Step-by-step explanation:
area of shape A = 3cm²
number of A shape in shape B = 9 and half
1 shape A = 3cm²
9 shape A = 3×9 = 27cm²
1 half shape A = 3 ÷ 2 = 1.5cm²
area of shape B = 27cm² + 1.5cm² = 28.5cm²
_____________________Follow mePlease answer thank you
Answers
Answer:
I think it's the 2nd one: (6,0) (0,-3)
7. The Taylor Rule states that the central bank should set the short-term nominal interest rate (i)
based on the inflation gap [the difference between inflation (3.14) and desired inflation (3.14*)] and the
output gap (the percentage difference between real GDP (Y) and potential GDP (Y*) An
example of a Taylor Rule would be the formula
i - 3.14 = 1.5 +0.5(3.14-3.14*) +0.5 (Y-Y*/Y*)
The term on the left-hand side is the real interest rate. Consider the following table
Inflation rate (3.14), %
Target inflation rate (3.14*), %
Output gap, %
Real interest rate
Nominal interest rate
Base Scenario Scenario B Scenario C
4.0
20
2.0
20
0.0
20
20
20
00
a. Fill in the real and nominal interest rates chosen by the policy maker in the base scenano
b. How does scenario B differ from the base scenario in terms of the inflation and output gaps?
Calculate the real interest rate. Has the real interest rate moved in the direction that would
move the inflation rate toward its target?
c. How does scenario C differ from the base scenario in terms of the inflation and output gaps?
Calculate the real interest rate. Has the real interest rate moved in the direction that would
move output toward the potential level?
d. Suppose a new chair of the central bank is appointed and she switches to a new policy rule of
the form given in the next equation. Recalculate the real and nominal interest rates for the
three scenarios. What has been the effect of the change in weights?
i-3.14=1.5 +0.75(3.14-3.14*) +0.25(Y-Y*/Y*)
Answers
The weight on the inflation gap has increased from 0.5 to 0.75. The real interest rate is 16.86% and Nominal interest rate is 20%
a. In the base scenario, the real interest rate will be 20%, and the nominal interest rate will be 20%.
b. In scenario B, inflation rate will be higher (4%) compared to the base scenario (3.14%).
Output gap is 0% in both the scenarios, however, in the base scenario inflation gap is 0% (3.14 - 3.14) and in scenario B, inflation gap is 0.86% (4 - 3.14).
Now, let's calculate the real interest rate.
Real interest rate in base scenario = 20% - 3.14%
= 16.86%.
Real interest rate in scenario B = 20% - 3.14% + 1.5 + 0.5 (4-3.14) + 0.5 (0-0/0)
= 19.22%.
The real interest rate has moved in the direction to move inflation rate towards its target.
c. In scenario C, the output gap will be 20% compared to 0% in the base scenario.
Inflation gap is 0% in both the scenarios
Inflation rate is 3.14% and in scenario C, inflation rate is 2%.
Let's calculate the real interest rate. Real interest rate in the base scenario = 20% - 3.14%
= 16.86%.
Real interest rate in scenario C = 20% - 2% + 1.5 + 0.5 (3.14 - 3.14) + 0.5 (20-0/20)
= 20.15%.
Real interest rate in scenario C = 20% - 2% + 1.5 + 0.75 (3.14-3.14) + 0.25 (20-0/20) = 18.78%.
The new policy rule has changed the weight of the output gap in the Taylor Rule from 0.5 to 0.25.
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Which statement best describes the formation of metamorphic rocks?
A. Magma cools and crystallizes slowly beneath Earth’s surface.
B. Liquid lava erupts from a volcano and cools quickly at Earth’s surface.
C. Sand and gravel are deposited at the mouth of a river and, over time, form layers that solidify into rock.
D. Ancient sandstone beaches become buried deep beneath Earth’s surface, where they are exposed to extreme heat and pressure.
Answers
Answer:
D
Step-by-step explanation:
Metamorphic rock is formed when rocks or minerals are subjected to high heat and high pressure, or a combination of these two factors. Answer D is the only answer that explicitly describes a rock or mineral being subject to these two things, so therefore D is the best answer.
hope this helps you :))
Solve the word problem. Complete Steps 1-2 in order.
The area of Bernie's porch can be represented by (x²-17x+60) ft².
1. Step 1: Find the dimensions of
the porch.
O (x-30) by (x+2) ft
O (x-5) by (x-12) ft
O (x+20) by (x-3) ft
O (x-20) by (x-3) ft
2. Step 1: Find the perimeter of
the room.
O(4x-34) ft
O (4x-56) ft
O (4x+34) ft
O (4x-46) ft
Answers
To find the dimensions of the porch, we need to factor the quadratic expression x² - 17x + 60 and then use the factors to determine the length and width of the porch. We can factor the expression as follows:
x² - 17x + 60 = (x - 12)(x - 5)
Therefore, the dimensions of the porch are:
Length = x - 5
Width = x - 12
So, the correct answer is:
O (x - 5) by (x - 12) ft
Step 2: Find the perimeter of the room.
To find the perimeter of the porch, we need to add up the lengths of all four sides. The length and width of the porch are x - 5 and x - 12, respectively, so the perimeter is:
2(x - 5) + 2(x - 12) = 4x - 34
Therefore, the correct answer is:
O (4x - 34) ft
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Answers
The sequence is decreasing as n increases and sequence converges to the value 0.
The given sequence is defined as aₙ = 1 / (7n + 3).
To determine if the sequence converges or diverges, we need to analyze its behavior as n approaches infinity.
As n increases, the denominator 7n + 3 also increases which means that the values of aₙ will get smaller and smaller, approaching zero as n becomes larger.
The sequence converges to the value 0.
The sequence is decreasing as n increases.
The sequence converges to the value 0.
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Which statement best describes the function?
A. The function is always increasing,
B. The function is never increasing
C. The function is increasing when x 0.
D. The function is increasing when x 0
Answers
Answer:
I dont see numbers of a function
Step-by-step explanation:
What is Set-builder and give example
Answers
It is just a way of describing a set.
You put the set in braces (sometimes called curly brackets), {}, which is a typical way to show that you are talking about sets.
Here are a few examples:
{x| x > 0} This is read: The set of all x such that x is greater than 0.
Reading it character by character:
{ — shows you that we are talking about a set, and it sort of opens up, or starts the set
x — tells you that we are talking about a set of elements x
| (and a : is also used sometimes) — such that (this is just notation). This says that you are about to be given to rule to see what “rule” or characteristic the x’s will have.
x > 0 — This just says what characteristics x must have to be in this set. So 7, 1/2, .3 and pi are all in this set, while -1, 0, -radical(2) are not.
So this example is just how you would use set builder notation to talk about the set “the positive real numbers”
So why use it? Because it can describe more complicated sets much more succinctly.
2. {3n + 1 | n is an element of the natural numbers} (and this could be written more succinctly if I used certain symbols) would be the set {4, 7, 10, 13, …}
3. {n^2 + n + 1 | n is an element of the natural numbers} would be the set {3, 7, 13, 21, …} and notice here, if you didn’t see the set builder notation, you might have trouble knowing what the next term in the set was.
a teacher figures that final grades in the statistics department are distributed as: a, 25%; b, 25%; c, 40%; d, 5%; f, 5%. at the end of a randomly selected semester, the following number of grades were recorded. find only the test statistics to test the claim that the grade distribution for the department is different than expected. be sure to set up the equation. do not conduct the hypothesis testing. grade a b c d f number 42 36 60 8 14
Answers
The critical value is 13.28 when the distribution of final marks in the statistics department.
Given that,
According to a teacher, the distribution of final marks in the statistics department is: A, 25%; B, 25%; C, 40%; D, 5%; and F, 5%. The following number of grades were recorded at the conclusion of a randomly chosen semester.
We have to recognize the important value.
We know that,
Given a=0.01,
The critical value of chi-square with 0.99,
df=n-1=4 is
13.28 (From chi-square table)
Therefore, The critical value is 13.28 when the distribution of final marks in the statistics department.
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DONT IGNORE PLEASE!! What is the measure of angle c
A. 94°
B. 86°
C. 18°
D. 274°
Answers
Answer:
B
Step-by-step explanation:
the sum of the 3 angles in a triangle is 180° , so
∠ c + 56° + 38° = 180°
∠ c + 94° = 180° ( subtract 94° from both sides )
∠ c = 86°
Point C has a coordinate of (-4, -6) and point D has a coordinate of (1, -6), how far are they apart?
Answers
The distance between points C and D is given as follows:
5 units.
How to calculate the distance between two points?Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates \((x_1,y_1)\) and \((x_2,y_2)\).
The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:
\(D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
The coordinates for this problem are given as follows:
(-4, -6) and (1, -6).
Hence the distance is given as follows:
\(D = \sqrt{(-4 - 1)^2 + (-6 - (-6))^2}\)
D = 5 units.
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How many solutions does the system of equations below have? y=-3/4x+1/6
Answers
The solution is the point (0, 1/6) y = 1/6
Given the equation y = (-3/4)x + 1/6, which represents a linear equation, there is no "system" of equations involved since there is only one equation.
In this case, the equation is in slope-intercept form (y = mx + b),
where m represents the slope (-3/4) and b represents the y-intercept (1/6).
The slope-intercept form allows us to determine various properties of the equation.
Since there is only one equation, the solution to this equation is a single point on the Cartesian plane.
Each pair of x and y values that satisfy the equation represents a solution.
For example, if we choose x = 0, we can substitute it into the equation to find the corresponding y value:
y = (-3/4)(0) + 1/6
y = 1/6
Therefore, the solution is the point (0, 1/6).
In summary, the given equation has a unique solution, represented by a single point on the Cartesian plane.
Any value of x plugged into the equation will yield a corresponding y value, resulting in a unique point that satisfies the equation.
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Solve for x: -5bracket(-3x+1)=-45
Answers
Step-by-step explanation:
-5(-3x+1)=-45
Expand bracket: 15x-5=-45
15x= -45+5
15x=-40
X=-2.67
I hope this helps.
CAN SOMBODY HELP ME IF YOU DO IT WRONG I WILL REPORT YOU BUT IF IT IS RIGHT I WILL GIVE YOU ALL MY POINTS AND BRINLY AND 5 STARS AND A THANKS BUT ALL I ASK IS THAT WHEN YOU ANSWER IT PLZZZZZZZZZ TELL ME HOW YOU DID IT I NEED TO NOW BECAUSE I NEED YOUR HELP
Answers
Answer:
D: 2:3
Step-by-step explanation:
She saw 2/5 mockingbirds
The remainder of birds she could see is 3
This means she saw 3 blue jays
The ratio is mockingbirds to blue jays, which is 2:3
During the year ? , spending reached $2800 per household
Answers
Solution:
The function that models average annual expenditures per household, in dollars.
\(f(x)=2577e^{0.0359x}\)Where x represents the year, where x = 0 corresponds to 2003.
\(\begin{gathered} x=0..........2003 \\ \\ x=1...........2004 \\ \\ x=2.............2005 \\ \\ x=3.............2006 \\ \\ x=4..............2007 \\ \\ x=5..............2008 \end{gathered}\)Thus;
\(\begin{gathered} f(5)=2577e^{0.0359(5)} \\ \\ f(5)=3083.69 \\ \\ f(5)\approx3084 \end{gathered}\)ANSWER: $3084
(b)
\(\begin{gathered} f(x)=2800 \\ \\ 2800=2577e^{0.0359x} \\ \\ \frac{2800}{2577}=\frac{2577e^{0.0359x}}{2577} \\ \\ 1.0865=e^{0.0359x} \\ \\ \text{ Take the logarithm of both sides;} \\ \\ \ln(1.0865)=\ln(e^{0.0359x}) \\ \\ 0.0830=0.0359x \\ \\ \frac{0.0830}{0.0359}=\frac{0.0359x}{0.0359} \\ \\ x=2.31 \\ \\ x\approx2 \end{gathered}\)The annual spending reached $2800 per household during the year 2005.
A relay race is 1/10 kilometer long. Four athletes will run an equal distance to complete the relay. How far does each athlete run?
Answers
Answer:1/40 kilometer
Step-by-step explanation:
Write the expression without using exponents.
(−9x)4
Answers
The expression (-9x)^4 can be represented as -6561x^3 without using exponents.
To express the expression (-9x)^4 without using exponents, we can expand it by multiplying the base (-9x) four times using the multiplication property.
(-9x)^4 = (-9x) * (-9x) * (-9x) * (-9x)
To simplify this expression, we can multiply the terms together, taking care to apply the rules of multiplication:
(-9x) * (-9x) = (-9 * -9) * (x * x) = 81 * x^2 = 81x^2
So, by substituting this result back into the original expression, we get:
(-9x)^4 = 81x^2 * (-9x) = -6561 x^3
Therefore, the expression (-9x)^4 can be represented as -6561x^3 without using exponents.
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HELP ASAP PLEASE :(!!!
given the following linear function sketch the graph of the function and find the domain and range. Upload your
document in the box below.
f(x) = -3x+7
Answers
Answer:
Step-by-step explanation:
You have the following function:
\(f(x)=-3x+7\)
The y-intercept of the function is given by the independent coefficient, which is 7.
y-intercept = 7
To obtain the x-intercept you equal the function to zero and solve for x, as follow:
\(0=-3x+7\\\\3x=7\\\\x=\frac{7}{3}\approx2.33\)
x-intercept = 2.33
Due to the coefficient of x is negative, the slope of the function is negative.
The function is a straight line, then, its domain all all real numbers and its range are all real numbers.
The graph of the function is attached in the image below.
17. Toby is riding his bicycle at 15 m/s. If it
takes him 60 seconds to get to the end of
the street. What was the length of the
street?
18.
met
him
Answers
Answer:
900 meters long
Step-by-step explanation:
Toby is riding his bicycle at 15 m/s. If it takes him 60 seconds to get to the end of the street. What was the length of the street?
at 15 meters per second for 60 seconds:
= time * speed
60* 15= 900 meters traveled,
so the street is 900 meters long.
18) What is the slope of the line that contains points (–6, –6) and (–3, 1)?
Answers
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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Leo is a great babysitter. He makes $8 an hour. Maureen is an even better babysitter. She makes $10 an hour. Maureen works 4 hours each week. How many hours does Leo need to work in order to earn as much money as Maureen earns in a week?
Answers
Answer:
5 hours
Step-by-step explanation:
put the steps, for changing the formula for sector area of a circle in degrees to the formula for the sector area of a circle in radians, in the correct order
Answers
Answer:
see explanation
Step-by-step explanation:
The area (A) of a sector is calculated as
A = area of circe × fraction of circle
= πr² × \(\frac{0}{360}\) ← where θ is in degrees
[ note that 360° = 2π radians ], then
A = πr² × \(\frac{0}{2\pi }\) ← where θ is in radians
( Cancel π on numerator/ denominator )
A = \(\frac{1}{2}\)θr²
9514 1404 393
Answer:
top down: 2, 4, 1, 7, 6, 5, 3, 8
Step-by-step explanation:
It appears the expected order may be ...
__
Write the formula for a sector area of a circle with central angle, θ, in degrees.
\(\textit{Area of a Sector}=\dfrac{\theta}{360^{\circ}}\cdot\pi r^2\)
Replace 360° with 2π radians.
\(\dfrac{\theta}{360^{\circ}}=\dfrac{\theta}{2\pi}\)
Replace the angle ratio in degrees with the angle ratio in radians.
\(\textit{Area of a Sector}=\dfrac{\theta}{2\pi}\cdot\pi r^2\)
Simplify by cancelling
\(\textit{Area of a Sector}=\dfrac{1}{2}\theta r^2\)