The LCM of the two polynomials is (x - 1)(x + 4)(x + 1).
Let's factorize the given polynomials:
Polynomial 1:
x² + 3x - 4 can be factored as (x - 1)(x + 4).
Polynomial 2:
2x² - 2 can be factored as 2(x - 1)(x + 1).
Now, we identify the highest power of each distinct factor:
The highest power of (x - 1) is (x - 1).
The highest power of (x + 4) is (x + 4).
The highest power of (x + 1) is (x + 1).
Therefore, the LCM of the two polynomials is (x - 1)(x + 4)(x + 1).
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DEFG models the shape of a pig pen on a farm. What is the perimeter of the pen, in units?
Answer:
Step-by-step explanation:
The price of a CD radio cassette has been reduced from $16,999 to $13,999 . Calculate the rate of discount
Answer:
The difference between the original price of $16,999 and the discounted price of $13,999 is:
$16,999 - $13,999 = $3,000
So the discount is $3,000.
To calculate the rate of discount as a percentage, you need to divide the discount by the original price, and then multiply by 100.
($3,000 / $16,999) x 100% ≈ 17.65%
Therefore, the rate of discount is approximately 17.65%.
Step-by-step explanation:
Answer:
0.17648 (round to whatever you need) is the RATE of discount
17.648% is the PERCENTAGE of discount
Step-by-step explanation:
(16,999 - 13,999)/ 16,999 = 0.17648
Because of the formula (og price - discount price)/ og price
Kevin has a rectangular prism that was made up of identical cubes, with no gaps between any cubes. The following figure shows the prism and one of the cubes. What was the total number of the cubes used to make up the prism?
The total number of cubes used to make up the rectangular prism can be determined by counting the number of cubes in each dimension and multiplying the values together.
To start, we need to determine the number of cubes in each dimension of the rectangular prism. Looking at the figure, we can see that the length of the rectangular prism is made up of 5 cubes, the width is made up of 3 cubes, and the height is made up of 4 cubes.
Multiplying these values together gives us:
5 x 3 x 4 = 60
Therefore, the total number of cubes used to make up the rectangular prism is 60.
In summary, to find the total number of cubes used to make up a rectangular prism, we need to count the number of cubes in each dimension and multiply them together. In this specific example, the rectangular prism was made up of 5 cubes in length, 3 cubes in width, and 4 cubes in height, resulting in a total of 60 cubes.
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ADCB has the following side measures. AD =84 cm, DC=28 cm, CB=5x+24 and AB = 3x-8. What is the value of x that makes ADCB a parallelogram
Step-by-step explanation:
In a parallelogram, opposite sides are equal in length. Therefore, we can set AD equal to CB and DC equal to AB to get two equations:
AD = CB
84 = 5x + 24
DC = AB
28 = 3x - 8
Solving the first equation for x, we get:
84 = 5x + 24
60 = 5x
x = 12
Solving the second equation for x, we get:
28 = 3x - 8
36 = 3x
x = 12
Since both equations give us x = 12, we can conclude that the value of x that makes ADCB a parallelogram is x = 12.
To verify, we can check that the opposite sides are equal in length:
AD = 84 cm
CB = 5x + 24 = 5(12) + 24 = 84 cm
DC = 28 cm
AB = 3x - 8 = 3(12) - 8 = 28 cm
Both pairs of opposite sides have the same length, so ADCB is a parallelogram when x = 12.
Answer:
x = 3.0
Step-by-step explanation:
This makes ADCB a parallelogram, with AD as its base and AB and CB as its two parallel sides.
Match the system of equations graphed to the correct solution.
infinate solutions (1,1) (-1,2) -1.5,4.5
The solution for each system of equations in this problem is given as follows:
First system: no solution.Second system: (1,1).Third system: (-1.5, 4.5).Fourth system: (-1, 2).How to solve a system of equations?Considering the graph containing the equations for the system, the solution of the system of equations is presented by the point of intersection of all the equations of the system.
When the two curves do not intersect, as is the case for the parallel lines in the first graph in this problem, it is said that the system has no solution.
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if a three-digit number is formed from the digits one through five, using any digit only once, what is the probability that the number will be divisible by 3? express your answer as a common fraction.
The probability that a three-digit number formed from the digits one through five, using any digit only once, will be divisible by 3 is $\frac{2}{5}$.
To find out how many three-digit numbers can be formed using the digits one through five, we first need to determine how many options we have for the first digit. Since the number cannot start with zero, there are four options: 1, 2, 3, or 4. For the second digit, there are only three options left since we cannot repeat the digit we used for the first digit. Finally, for the last digit, there are only two options left. So, in total, there are 4 x 3 x 2 = 24 three-digit numbers that can be formed using the digits one through five.
Now we need to figure out how many of these numbers are divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3. The only way to get a sum that is divisible by 3 using the digits one through five is to use three of them (1, 2, 3, 4, or 5). So we need to count the number of three-digit numbers that can be formed using three of the digits 1, 2, 3, 4, and 5. There are 5 ways to choose the digit that is left out, and then 3 x 2 = 6 ways to arrange the other three digits. So there are 5 x 6 = 30 three-digit numbers that can be formed using three of the digits 1, 2, 3, 4, and 5.
Therefore, the probability of selecting a three-digit number that is divisible by 3 is $\frac{30}{24} = \frac{5}{4}$. However, we need to simplify this fraction, so we divide the numerator and denominator by 5 to get $\frac{6}{5}$. Finally, we subtract this fraction from 1 to find the probability that a three-digit number formed from the digits one through five, using any digit only once, will not be divisible by 3: $1 - \frac{6}{5} = \boxed{\frac{2}{5}}$.
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3. SHORT ANSWER Which point on the number line
is closest to the product of the numbers graphed at
points R and T? Explain your answer.
P
QRST
0
1
The closest point will be 0.6 which is point Q.
To determine the product of the numbers graphed at points R and T, we need to find the coordinates of these points on the number line.
So, point R x point T
= 7/10 x 9/10
However, once you have the coordinates of points R and T, you can multiply the corresponding numbers to find their product.
Then, 0.7 x 0.9
= 0.63
From the points given we can write as
P = 0.4
Q = 0.6
R= 0.7
S = 0.8
T = 0.9
Among all the points the most closest is 0.6 which is point Q.
Thus, the closest point will be 0.6 which is point Q.
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Tomatoes to total number of carrots and cauliflower tomatoes 19 carat 12 curly flower 15 solve
The total number of carrots and cauliflower is 27, and the number of tomatoes is 19.
Explanation:
Let the number of carrots be c and the number of cauliflower be f.
From the given information, we know that:
c + f = 27 ----(1) (total number of carrots and cauliflower)
19 = tomatoes
We are asked to find the values of c and f. To do this, we can use the equation (1) and substitute 19 for f:
c + 19 = 27
c = 27 - 19
c = 8
Therefore, the number of carrots is 8. To find the number of cauliflower, we can use the equation (1) and substitute the value of c:
8 + f = 27
f = 27 - 8
f = 19
Therefore, the number of cauliflower is 19. Hence, the total number of carrots and cauliflower is:
c + f = 8 + 19 = 27
And the number of tomatoes is:
tomatoes = 19
So, the given information can be summarized as 8 carrots, 19 cauliflower, and 19 tomatoes.
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Point A has coordinates (-13, -9).
Point B has coordinates (1, 15).
What are the coordinates of the midpoint of
line AB?
Answer:(-6,3)
Step-by-step explanation:
helpppppp i need help with this rn
A map of an amusement park is shown on the coordinate plane with the approximate location of several rides.
coordinate plane with points at negative 14 comma 1 labeled Woozy Wheel, negative 6 comma 2 labeled Bumper Boats, negative 2 comma negative 4 labeled Roller Rail, negative 2 comma negative 6 labeled Trolley Train, 2 comma negative 3 labeled Silly Slide, and 6 comma 11 labeled Parachute Plunge
Determine the distance between the Woozy Wheel and the Roller Rail.
119 units
11 units
169 units
13 units
VIDEO GAMES Soo Jin is designing a character for a video game using the items in the table. If she selects one item from each of the four categories, how many different possible characters can Soo Jin create?
It's important to note that this calculation assumes that the items in each category are distinct and that Soo Jin can select any item from each category without any restrictions or dependencies.
To determine the number of different possible characters Soo Jin can create for the video game, we need to consider the combinations of items from each category.
Let's assume there are n1 items in Category 1, n2 items in Category 2, n3 items in Category 3, and n4 items in Category 4. To find the total number of possible characters, we multiply the number of options in each category:
Total number of characters = n1 * n2 * n3 * n4
For example, if Category 1 has 5 items, Category 2 has 3 items, Category 3 has 2 items, and Category 4 has 4 items, the total number of possible characters would be:
Total number of characters = 5 * 3 * 2 * 4 = 120
So Soo Jin can create 120 different characters by selecting one item from each of the four categories.
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I need your Help please this is due on wen but needs to be turned in by 12am or i get a F so i need help with this please and thank you!
Answer: A (300 in^2)
Step-by-step explanation: You can either use volume or the area to find out how much wrapping paper Malia will need. so, I will use the area so I took 2in and mutipyied it by 10in which got me 20in then I took the 20in I got from 2in mutiplied by the 10in and mutipled it by 15 and got the answer of 300 in^2. so the answer is a .
The length of a rectangle room is 5m and the width is 4m find the perimeter of the room
Answer:
18m
Step-by-step explanation:
Perimeter = 2L + 2W
Perimeter = 2(5) + 2(4)
= 10 + 8
= 18 m
Answer:
18 meters
Step-by-step explanation:
5+5=10
4+4=8
10+8=18
ony is given the following quiz:
Round $5/11 = 0.\overline{45}$ to the nearest
1) whole number
2) tenth
3) hundredth
4) thousandth
5) ten thousandth
6) hundred thousandth
Tony completes the quiz as follows. He first (correctly) rounds $0.\overline{45}$ to the nearest hundred thousandth, and writes it as his answer for question $6$. Then he rounds his answer for question $6$ to the nearest ten thousandth and uses that as his answer for question $5$. Then he rounds his answer for question $5$ to the nearest thousandth and uses that as his answer for question $4$. Then he rounds his answer for question $4$ to the nearest hundredth and uses that as his answer for question $3$. Then he rounds his answer for question $3$ to the nearest tenth and uses that as his answer for question $2$. Finally, he rounds his answer for question $2$ to the nearest whole number, uses that as his answer for question $1$, and turns in the quiz.
Step-by-step explanation:
1. To round $5/11 = 0.\overline{45}$ to the nearest whole number, we look at the digit in the ones place, which is 5. Since 5 is greater than or equal to 5, we round up to the next whole number, which is 1. Therefore, rounded to the nearest whole number, $5/11$ is equal to 1.
2. To round $5/11 = 0.\overline{45}$ to the nearest tenth, we look at the digit in the hundredth place, which is 5. Since 5 is greater than or equal to 5, we round up to the next tenth, which is 0.5. Therefore, rounded to the nearest tenth, $5/11$ is equal to 0.5.
3. To round $5/11 = 0.\overline{45}$ to the nearest hundredth, we look at the digit in the thousandth place, which is also 5. Since 5 is greater than or equal to 5, we round up to the next hundredth, which is 0.46. Therefore, rounded to the nearest hundredth, $5/11$ is equal to 0.46.
4. To round $5/11 = 0.\overline{45}$ to the nearest thousandth, we look at the digit in the ten thousandth place, which is 4. Since 4 is less than 5, we round down to 0.455. Therefore, rounded to the nearest thousandth, $5/11$ is equal to 0.455.
5. To round $5/11 = 0.\overline{45}$ to the nearest ten thousandth, we look at the digit in the hundred thousandth place, which is 5. Since 5 is equal to 5, we round up if the digit in the ten thousandth place is odd, and round down if it is even. In this case, the digit in the ten thousandth place is 4, which is even, so we round down to 0.4550. Therefore, rounded to the nearest ten thousandth, $5/11$ is equal to 0.4550.
6. To round $5/11 = 0.\overline{45}$ to the nearest hundred thousandth, we look at the digit in the millionth place, which is also 5. Since 5 is equal to 5, we round up if the digit in the hundred thousandth place is odd, and round down if it is even. In this case, the digit in the hundred thousandth place is even, so we round down to 0.45499. Therefore, rounded to the nearest hundred thousandth, $5/11$ is equal to 0.45499.
When Tony rounds $0.\overline{45}$ to the nearest hundred thousandth, he gets 0.45499, which he then rounds to 0.4550, and so on, following the same steps we did above. His final answer for question 1 will be 0, since rounding 0.5 to the nearest whole number gives 0. Therefore, Tony's final answers will be:
1. 0
2. 0
3. 0.46
4. 0.455
5. 0.4550
6. 0.45499
11. How many combinations of four can you make from the set?
25 baseball cards
Put your answer in the form [XXXXX]. Please do not insert a comma.
A 30-foot-long support wire for a 16-foot high streetlight runs from the top corner of a building to a point on the ground, forming a straight line. The length of the wire from the top of the building to the top of the street light is 6 feet. How tall is the building?
A. 16 feet
B.20 feet
C. 32 feet
D.48 feet
Answer:
B. 20 feet
Step-by-step explanation:
(30 - 6) = 30 = 16 = x
24 = 30 = 16 = x
4 = 5 = 16 = x
4x = 80
x = 20
NEED HELP ASAP A garden is 6 yards long. How long is the garden in meters? There are
approximately 11 meters in 12 yards. Round your answer to the
nearest tenth.
The garden is approximately 5.5 meters long.
1 yard is approximately 0.91 meters (since 1 meter is approximately 1.094 yards). Therefore, 6 yards is approximately 5.5 meters (6 x 0.91).
Proportionality refers to the relationship between two quantities that change together in a consistent manner. In a proportional relationship, when one quantity changes, the other quantity changes in direct proportion to it.
Since 11 meters is approximately 12 yards, we can set up a proportion:
6 yards = x meters
12 yards = 11 meters
Solving for x, we get:
x = (6 yards)(11 meters/12 yards) = 5.5 meters
Rounding to the nearest tenth, we get that the garden is approximately 5.5 meters long.
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find and sketch the domain of the function. f(x, y) = ln(4 − x2 − 4y2)
The domain of the function is the set of all points in the xy-plane that satisfy the inequality 4 - x^2 - 4y^2 > 0.
The domain of a function consists of all the possible input values that can be plugged into the function to produce a valid output. In the case of the function f(x,y) = ln(4 - x^2 - 4y^2), the argument of the natural logarithm must be positive for the function to be defined. Therefore, we need to find the set of (x,y) pairs that satisfy the inequality:
4 - x^2 - 4y^2 > 0
Rearranging this inequality, we get:
x^2 + 4y^2 < 4
This is the equation of an ellipse centered at the origin with semi-major axis of length 2 and semi-minor axis of length 1/2. Thus, the domain of the function f(x,y) consists of all the points inside this ellipse.
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(T/F) if two non-zero vectors a and b satisfy proja with arrowb = 0 then a and b are parallel.
The statement, "two non-zero vectors "a" and "b" satisfy a×b = 0 then "a" and "b" are parallel" is True, because the cross-product of two vectors is always 0.
The "Cross-Product" of "two-vectors" is zero only when the vectors are parallel or when one (or both) of the vectors is the zero vector. Since a × b = 0 and both "a" and "b" are non-zero, it implies that "a" and "b" are parallel. This is because the cross product of parallel vectors is always zero.
Therefore, the condition a × b = 0 serves as a test for parallelism between non-zero vectors, so the statement is True.
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The given question is incomplete, the complete question is
(T/F) If two non-zero vectors "a" and "b" satisfy a×b = 0 then "a" and "b" are parallel.
let A and B be a square matrices of order 3 defined By A={aij}=2i^2+3j and B={bij}=i^2+2j^2. Compute matrices A and B
(b) find the sum of their determinant
(c) 2A^-1
Matric A = [[5, 8, 11],
[11, 14, 17],
[21, 24, 27]]
Matric B = [[3, 9, 19],
[6, 12, 22],
[11, 17, 27]]
(b) The sum of their determinant is 102.
(c) The inverse of matrix A, you can use matrix inversion techniques such as the adjugate matrix or row reduction.
To compute the matrices A and B, we substitute the given expressions for each element into their respective matrices.
Matrix A:
A = [aij] where aij = 2i² + 3j
Substituting the values of i and j from 1 to 3, we get:
A = [[2(1)² + 3(1), 2(1)² + 3(2), 2(1)² + 3(3)],
[2(2)² + 3(1), 2(2)² + 3(2), 2(2)² + 3(3)],
[2(3)² + 3(1), 2(3)² + 3(2), 2(3)² + 3(3)]]
Simplifying each element:
A = [[2 + 3, 2 + 6, 2 + 9],
[8 + 3, 8 + 6, 8 + 9],
[18 + 3, 18 + 6, 18 + 9]]
A = [[5, 8, 11],
[11, 14, 17],
[21, 24, 27]]
Matrix B:
B = [bij] where bij = i² + 2j²
Substituting the values of i and j from 1 to 3, we get:
B = [[1² + 2(1)², 1² + 2(2)², 1² + 2(3)²],
[2² + 2(1)², 2² + 2(2)², 2² + 2(3)²],
[3² + 2(1)², 3² + 2(2)², 3² + 2(3)²]]
Simplifying each element:
B = [[1 + 2, 1 + 8, 1 + 18],
[4 + 2, 4 + 8, 4 + 18],
[9 + 2, 9 + 8, 9 + 18]]
Next, we move on to the remaining parts of the question.
To find the sum of the determinants of matrices A and B, we compute the determinants separately and add them together:
det(A) = |A|
= 5(14 - 17) - 8(11 - 17) + 11(11 - 14)
= -6
det(B) = |B| = 3(12 - 17) - 9(6 - 17) + 19(6 - 12)
= 108
Sum of determinants = det(A) + det(B)
= -6 + 108
= 102
To compute 2A⁻¹, we need to find the inverse of matrix A and multiply it by 2:
A⁻¹ = inverse of matrix A
Once you have the inverse, you can multiply it by 2:
2A^-1 = 2 × A⁻¹
Since matrix A is not given in the question and we've only computed its elements based on the given formula
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PLEASE HELP!!
The residuals for data set A and data set B were calculated and plotted on separate residual plots. If the residuals for data set A do not form a pattern, and the residuals for data set B form a pattern, what can be concluded?
A. Data set A is linear, and data set B is not linear.
B. Data set A is not linear, and data set B is not linear.
C. Data set A is linear, and data set B is linear.
D. Data set A is not linear, and data set B is linear.
Answer: A
Step-by-step explanation:
When the residuals are random, it implies that the linear fit is appropriate for the data set. When residuals form a pattern, it indicates that the linear fit for the data set is not appropriate since there is a systematic error instead of random error.
can someone give me an example on how to do these and how to set it up?
The numeric values for the combinations of function are given as follows:
(f + g)(3) = 9.(f x g)(-1) = -8.(f ∘ g)(2) = 8.(g ∘ f)(0) = 2.(g ∘ g)(3) = -1.How to obtain the numeric values of the function?The numeric values for the function are obtained applying the operations from the table.
For the addition, we just add the numeric values, hence:
(f + g)(3) = f(3) + g(3) = 8 + 1 = 9.
For the multiplication, we just multiply the values, hence:
(f x g)(-1) = f(-1) x g(-1) = -2 x 4 = -8.
For the composition, the output of the inner function is the input to the outer function, hence:
(f ∘ g)(2) = f(g(2)) = f(3) = 8.(g ∘ f)(0) = g(f(0)) = g(0) = 2.(g ∘ g)(3) = g(g(3)) = g(1) = -1.Learn more about the numeric values of a function at brainly.com/question/28367050
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The points C(−6,−1), D(−4,5), and E(2,3) form a triangle. Plot the points then click the "Graph Triangle" button.
The slope of CD = 0.6.
The slope of DE = -1/3.
The slope of EC = 0.5.
The length of CD = √136 units.
The length of DE = √40 units.
The length of EC = √80 units.
Triangle CDE is a scalene triangle.
How to calculate or determine the slope of a line?In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = rise/run
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) CD = (5 + 1)/(4 + 6)
Slope (m) CD = 6/10
Slope (m) CD = 0.6.
Slope (m) DE = (3 - 5)/(2 + 4)
Slope (m) DE = -2/6
Slope (m) DE = -1/3.
Slope (m) EC = (3 + 1)/(2 + 6)
Slope (m) EC = 4/8
Slope (m) EC = 0.5.
Next, we would determine the length of each sides of triangle CDE as follows;
Distance CD = √[(x₂ - x₁)² + (y₂ - y₁)²]
Distance CD = √[(4 + 6)² + (5 + 1)²]
Distance CD = √136 units.
Distance DE = √[(2 + 4)² + (3 - 5)²]
Distance DE = √40 units.
Distance EC = √[(2 + 6)² + (3 + 1)²]
Distance EC = √80 units.
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An outline of a city in Florida, with measurements in miles (mi) is shown.
0.75 miles
In 2019, there were 60, 500 people in this town. Which of the following is the approximate population density of the city?
A 60 people/mi²
The approximate population density of the city is 1,005 people/mi²
The correct answer choice is option D.
What is population density?Population density can be defined as the number of people in a geographic location at a particular time. It is calculated by dividing the total area of a region by the total number of people that live in that area.
Number of square grids = 107
Land mass = 0.75 miles
Number of people = 60, 500
One square grid = 0.75 × 0.75
= 0.5625 square miles
Total area = Area × Number of square grids
= 0.5625 square miles × 107
= 60.1875 square miles
Population density = Number of people / Total area
= 60,500 /60.1875 square miles
= 1005.192107995846
Approximately,
1,005 people/mi²
Hence, 1,005 people/mi² is the population density of the city.
Complete question:
A. 60 people/mi²
B. 565 people/mi²
C. 754 people/mi²
D. 1,005 people/mi²
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Draw a picture that shows both the ratio 4:11 and 7:11
The picture showing the ratios 4:11 and 7:11 is attached accordingly.
what is ratio and why is it important?Ratios are common in everyday life and assist to simplify many of our interactions by putting numbers into context. Ratios facilitate the measurement and expression of numbers by making them more understandable. Ratios in everyday life: The automobile was moving at 60 mph, or 60 miles per hour.
A ratio is a comparison expressed as a quotient of two numbers, a and b, and is commonly represented as ab or a/b.
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A jar full of marbles is displayed the following table shows the guesses for 10 people. The actual number of marbles in the jar is 145. Calculate the absolute guessing error for all 10 guesses.
Answer:
Step-by-step explanation:
To calculate the absolute guessing error, we need to find the absolute difference between each guess and the actual number of marbles (145).
Guess Absolute Guessing Error
190 45
150 5
125 20
133 12
167 22
160 15
148 3
200 55
170 25
115 30
To check, we can verify that each absolute guessing error is the positive difference between the guess and 145.
In the diagram below, CD is a tangent to the circle centre O. If ABC = 54°, calculate the size of C₁.
Answer:
the size of C1=54°[converse of a tan chord theorem]
Step-by-step explanation:
since the tan chord theorem state that the angle between a tangent and a chord is equal to the angle sub tended by that chord and in this situation your given the sub tend angle we intend to say is a converse of the tan chord
(2) There is a total of 20 balls. Some balls weigh 50g each and others weigh 120g each. If the 20 balls weigh 1560g altogether, how many 50g balls and 120g balls are there respectively?
Number of 50 g weigh balls = 12
And, Number of 120g weigh balls = 8
We have to given that;
There is a total of 20 balls. Some balls weigh 50g each and others weigh 120g each.
Let number of 50 g weigh balls = x
And, Number of 120g weigh balls = y
Hence, We can formulate;
x + y = 20 .. (i)
And, 50x + 120y = 1560
⇒ 5x + 12y = 156 .. (ii)
From (i);
x = 20 - y
Plug in (ii);
5 (20 - y) + 12y = 156
100 - 5y + 12y = 156
7y = 56
y = 8
From (i);
x = 20 - y
x = 20 - 8
x = 12
Thus, number of 50 g weigh balls = 12
And, Number of 120g weigh balls = 8
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How do I solve this?
Answer:
sec θ csc θ
Step-by-step explanation:
identities: sin²θ + cos²θ = 1, sec θ = 1/cos θ, csc θ = 1/sin θ.
secθ csc θ = (1/cosθ) (1/sinθ) = 1/(cosθsinθ) = 1/(sinθcosθ)
tan θ + cot θ = sinθ/cosθ + 1/tan θ
= sin θ/cos θ + cosθ/sinθ
= (sin²θ + cos²θ) / (sinθ cosθ)
= 1/sinθcosθ
= 1/cosθsinθ
= sec θ csc θ
