The given differential equation has characteristic equation
[tex]r^5 - 4r^4 + 4r^3 - r^2 + 4r - 4 = 0[/tex]
Solve for the roots [tex]r[/tex].
[tex]r^3 (r^2 - 4r + 4) - (r^2 - 4r + 4) = 0[/tex]
[tex](r^3 - 1) (r^2 - 4r + 4) = 0[/tex]
[tex](r^3 - 1) (r - 2)^2 = 0[/tex]
[tex]r^3 - 1 = 0 \text{ or } (r-2)^2=0[/tex]
The first case has the three cubic roots of 1 as its roots,
[tex]r^3 = 1 = 1e^{i0} \implies r = 1^{1/3} e^{i(0+2\pi k)/3} \text{ for } k\in\{0,1,2\} \\\\ \implies r = 1e^{i0} = 1 \text{ or } r = 1e^{i2\pi/3} = -\dfrac{1+i\sqrt3}2 \text{ or } r = 1e^{i4\pi/3} = -\dfrac{1-i\sqrt3}2[/tex]
while the other case has a repeated root of
[tex](r-2)^2 = 0 \implies r = 2[/tex]
Hence the characteristic solution to the ODE is
[tex]y_c = C_1 e^x + C_2 e^{-(1+i\sqrt3)/2\,x} + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
Using Euler's identity
[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]
we can reduce the complex exponential terms to
[tex]e^{-(1\pm i\sqrt3)/2\,x} = e^{-x/2} \left(\cos\left(\dfrac{\sqrt3}2x\right) \pm i \sin\left(\dfrac{\sqrt3}2x\right)\right)[/tex]
and thus simplify [tex]y_c[/tex] to
[tex]y_c = C_1 e^x + C_2 e^{-x/2} \cos\left(\dfrac{\sqrt3}2x\right) + C_3 e^{-x/2} \sin\left(\dfrac{\sqrt3}2x\right) \\ ~~~~~~~~ + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
For the non-homogeneous ODE, consider the constant particular solution
[tex]y_p = A[/tex]
whose derivatives all vanish. Substituting this into the ODE gives
[tex]-4A = 69 \implies A = -\dfrac{69}4[/tex]
and so the general solution to the ODE is
[tex]y = -\dfrac{69}4 + C_1 e^x + C_2 e^{-x/2} \cos\left(\dfrac{\sqrt3}2x\right) + C_3 e^{-x/2} \sin\left(\dfrac{\sqrt3}2x\right) \\ ~~~~~~~~ + C_3 e^{-(1-i\sqrt3)/2\,x} + C_4e^{2x} + C_5xe^{2x}[/tex]
Rewrite 1/2 x 1/2 x 1/2 as an exponential expression with a base of 2.
The equivalent expression of 1/2 * 1/2 *1/2 is 2^-3
How to rewrite the expression?The expression is given as:
1/2 * 1/2 *1/2
Apply the product law of indices
1/2 * 1/2 *1/2 = (1/2)^3
Apply the power law of indices
1/2 * 1/2 *1/2 = (2^-1)^3
This gives
1/2 * 1/2 *1/2 = 2^-3
Hence, the equivalent expression of 1/2 * 1/2 *1/2 is 2^-3
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need help with this asap
The converse, inverse and contrapositive of each conditional statement include:
Converse: If two angles have a common side, then they are adjacent.Inverse: If two angles are not adjacent, then they do not have a common side.Contrapositive: If two angles do not have a common side, then they are not adjacent.What is a conditional statement?
A conditional statement can be defined as a type of statement that can be written to have both a hypothesis and conclusion. Thus, it typically has the form "if P then Q."
Where:
P and Q represent sentences.
In this scenario, we would write the converse, inverse and contrapositive of each conditional statement as follows:
Converse: If two angles have a common side, then they are adjacent.Inverse: If two angles are not adjacent, then they do not have a common side.Contrapositive: If two angles do not have a common side, then they are not adjacent.Read more on conditional statement here: brainly.com/question/16951916
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If a boat with a value of $23, 000 depreciates at a rate of 18% per year, what is the value
after 5 years?
The value of the car after 5 years is 8527.02 dollars
How to find the value after depreciation?Depreciation is the reduction in the value of an asset over time due to elements such as wear and tear. For example a car is said to "depreciate" in value after a discovery of a faulty transmission.
The value after 5 years can be found as follows:
Rate = 18% = 18 / 100 = 0.18
Initial Value = $23,000
time = 5 years
Therefore,
y = 23000(1 + 18%)⁵
y = 23000(1 - 0.18)⁵
y = 23000(0.82)⁵
y = 23000 × 0.3707398432
y = 8527.0163936
y = 8527.02 dollars
Therefore, the value of the car after 5 years is 8527.02 dollars
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Using an exponential function, it is found that the value of the car after 5 years will be of $8,527.
What is an exponential function?A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
A(0) is the initial value.r is the decay rate, as a decimal.For this problem, the initial cost and the decay rate are given as follows:
A(0) = 23000, r = 0.18.
Hence the value of the boat after t years is given as follows:
[tex]A(t) = A(0)(1 - r)^t[/tex]
[tex]A(t) = 23000(1 - 0.18)^t[/tex]
[tex]A(t) = 23000(0.82)^t[/tex]
The value after 5 years will be of:
[tex]A(5) = 23000(0.82)^5 = 8527[/tex]
The value of the car after 5 years will be of $8,527.
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a coffee shop, the first 100 customers' orders were as follows. Small Medium Large Hot 5 48 22 Cold 8 12 5 What is the probability that a customer ordered a large given that he or she ordered a cold drink? Rounded to the nearest percent, [? ]% Enter
Using it's concept, there is a 20% probability that a customer ordered a large given that he or she ordered a cold drink.
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
In this problem, 8 + 12 + 5 = 25 people ordered a cold drink, and of them, 5 were large, hence the probability is:
p = 5/25 = 0.2 = 20%.
20% probability that a customer ordered a large given that he or she ordered a cold drink.
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Triangle S R Q is shown. Angle S R Q is a right angle. An altitude is drawn from point R to point T on side S Q to form a right angle. The length of T Q is 16 and the length of R Q is 20.
What is the length of Line segment S R?
9 units
12 units
15 units
18 units
The length of the line segment SR is 15 units.
How to find the length of a line segment?In Δ SRQ as seen in the attached image, we see that;
∠SRQ is a right angle
SQ is the hypotenuse
RT ⊥ SQ
Thus, by triangular rules, we know that;
(RQ)² = TQ × SQ
RQ = 20 units and TQ = 16 units
Thus;
(20)² = 16 × SQ
400 = 16 × SQ
SQ = 400/16
SQ = 25 units
By using Pythagoras theorem in Δ SRQ, we have;
(SR)² + (RQ)² = (SQ)²
RQ = 20 units and SQ = 25 units
Thus;
(SR)² + (20)² = (25)²
(SR)² + 400 = 625
(SR)² = 225
SR = √225
SR = 15 units
The length of the line segment SR is 15 units.
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Given the functions f(x) = x3 + x2 – 3x + 4 and g(x) = 2^x– 4, (Consider domain, range, x-intercepts, and y-intercepts.)
The first function is a type of cubic function and the second function is a type of linear function. The key feature(s) do f(x) and g(x) have in common as they both have value of domain and range as (-∞, ∞).
What is a Function?A function assigns the value of each element of one set to the other specific element of another set.
The function given in the problem is:
f(x) = x³ + x² – 3x + 4
The highest degree (power) of this function is 3. Thus, this function is a type of cubic function.
The second function given in the problem is:
g(x) = 2x – 4
The highest degree (power) of this function is 1. Thus, this function is a type of linear function.
Both the function has the value of domain and range from -∞ to ∞ (-∞,∞).
Thus, the first function is a type of cubic function and second function is a type of linear function. The key feature(s) do f(x) and g(x) have in common as they both have value of domain and range as (-∞, ∞).
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A procedure has an 81% success rate. The procedure is tried on 12 patients what is the probability that at least 8 patients show improvement
The probability that at least 8 patients show improvement is; 0.94
How to use Binomial Probability Distribution?The formula for Binomial Probability Distribution is;
P(X = r) = nCr * p^(r) * (1 - p)^(n - r)
where;
n = number of trials
r = number of specific events you wish to obtain
p is probability of success
From the question, we have;
n = 12
r = 8
p = 81% = 0.81
The probability that at least 8 patients show improvement is expressed as;
P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
Using online binomial probability calculator gives us;
P(X ≥ 8) = 0.11954 + 0.22649 + 0.28967 + 0.22453 + 0.07977
P(X ≥ 8) = 0.94
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f f(x) = 3x + 2 and g(x) = x2 + 1, which expression is equivalent to (f circle g) (x)?
hi
f°g mean apply g(x) and then use result of g(x) as "x" in f(x)
so as g(x) = x² +1 and f(x) = 3x+2
so if we call g(x) X we have
f( X) = 3 X +2
now we replace " X" for it's value
f(X) = 3 ( x² +1) +2
f(X) = 3x² +3 +2
f(X) = 3x² +5
so f°g = 3x² +5
(02.04 LC)
Determine the equation of the line shown in the graph:
graph of a vertical line with an x intercept of negative 1
y = −1
y = 0
x = −1
x = 0
Answer:
x = - 1
Step-by-step explanation:
the equation of a vertical line is
x = c
where c is the value of the x- coordinates the line passes through
the line passes through the x- intercept of - 1 , then
x = - 1 ← equation of vertical line
An H0-scale model railroad engine is 9 inches long. The H0 scale is 87 feet to 1 foot. How long is a real engine?
The real engine is 65.25 feet long
How to determine the length?The scale is given as:
Scale = 87 feet to 1 foot
The scale length is given as
Scale length = 9 inches
Convert inches to feet
Scale length = 9/12 feet
The actual length is then calculated as:
Actual = 87 * 9/12 feet
Evaluate
Actual = 65.25 feet
Hence, the real engine is 65.25 feet long
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6x+2y=8
9x+3y=14
Skew
Neither
Perpendicular
Parallel
4/3*(1/2-6)=(4/3*1/2)-(4/3-6)
Answer:
The product would be: -22/3 = 16/3, if true or false: false the left side -7.3 does not equal to the right side 5.3 which means the given statement is false, if it is evaluated -22/3 = 16/3, if multiplied it would still be -22/3 = 16/3 (you never said how to solve it so here are ways)
Lindsay is checking out books at the library, and she is primarily interested in mysteries and nonfiction. She has narrowed her selections down to six mysteries and nine nonfiction books. If she randomly chooses three books from her selections, what’s the probability that they will all be nonfiction? Enter a fraction or round your answer to 4 decimal places, if necessary.
Please explain in detail
The probability that all 3 books would be nonfiction is 0.1846.
What is the probability?Probability determines the chances that an event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
The probability that all 3 books would be nonfiction = (number of non fiction books / total number of books) x (number of non fiction books - 1 / total number of books - 1) x (number of non fiction books - 2 / total number of books - 2)
9 /15 x 8/14 x 7/13 = 0.1846
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If you go out to eat with 3 friends and your meal was $72.50, there is 6.75% sales tax
and you should tip the waiter 15%. How much should each person pay?
Tax $
Total meal cost
Tip $
Total cost of meal with tip
Price for each person
After the tax and the tip are counted, we conclude that each friend must pay $30.
How much should each person pay?We know that the meal cost is $72.50.
Now, there is a 6.75% sales tax and a 15% tip on the waiter (we assume that the tip is based on the total meal cost, after the tax is applied).
Then the total meal cost is:
C = $72.50*( 1 + 6.75%/100%) = $72.50*(1 + 0.0675) = $77.39
Now we also need to add the tip, which is a 15% of that, so the total cost with the tip included is:
C' = $77.39*(1 + 15%/100%) = $88.998 = $90
(Where we rounded up).
Then, if there are 3 friends, each one must pay:
$90/3 = $30
Each friend must pay $30.
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An animal shelter has a total of 5 * 47 animals comprised of cats, dogs, and rabbits. If the number of rabbits is 5 less than one half the number of cats, and there are 20 more cats than dogs, how many dogs are at the shelter?
The number of dogs in the animal shelter which comprise of cats, dogs, and rabbits is 84 dogs.
EquationTotal animals = 5 × 47
= 235
Number of rabbits = 1/2(d + 20) - 5Number of cats = d + 20Number of dogs = d235 = d + (d + 20) + 1/2(d + 20) - 5
235 = d + d + 20 + 1/2d + 10 - 5
235 = 2 1/2d + 25
235 - 25 = 2 1/2d
210 = 2 1/2d
d = 210 ÷ 2 1/2
d = 210 ÷ 5/2
= 210 × 2/5
= 420/5
d = 84 dogs
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Which of the following functions are discontinuous?
Answer:
D. I, II, and III
Step-by-step explanation:
A discontinuous function is a function which is not continuous.
If f(x) is not continuous at x = a, then f(x) is said to be discontinuous at this point.
To prove whether a function is discontinuous, find where it is undefined.
A rational function is undefined when the denominator is equal to zero.
Therefore, to find the values that make a rational function undefined, set the denominator to zero and solve.
Function I
Denominator: x - 2
Set to zero: x - 2 = 0
Solve: x = 2
Therefore, this function is undefined when x = 2 and so the function is discontinuous.
Function II
Denominator: 4x²
Set to zero: 4x² = 0
Solve: x = 0
Therefore, this function is undefined when x = 0 and so the function is discontinuous.
Function III
Denominator: x² + 3x + 2
Set to zero: x² + 3x + 2 = 0
Solve:
⇒ x² + 3x + 2 = 0
⇒ x² + x + 2x + 2 = 0
⇒ x(x + 1) + 2(x + 1) = 0
⇒ (x + 2)(x + 1) = 0
⇒ x = -2, x = -1
Therefore, this function is undefined when x = -2 and x = -1, and so the function is discontinuous.
Therefore, all three given functions are discontinuous.
I need helping solving this :)
Answer: answer is B
Step-by-step explanation:
use Pythagorean theorem to find other side. And sine is opposite side over hypotenuse.
You are training your dog to catch a frisbee. You are playing in a large field, and you are standing next to your dog when you throw the frisbee. If the path of the frisbee is y = –x2 + 9x + 8 and the path of the dog is modeled by y = 3x + 8, will the dog catch the frisbee? If so, what are the coordinates of the point or points where they meet?
Yes, they intersect at the coordinates (0, 8) and (3,17)
Yes, they intersect at the coordinates (0, 8) and (4, 20)
Yes, they intersect at the coordinates (0, 8) and (6, 26)
No, the paths do not cross
Solving a quadratic equation, the correct option is given by:
Yes, they intersect at the coordinates (0, 8) and (6, 26).
What is a quadratic function?A quadratic function is given according to the following rule:
y = ax² + bx + c
The solutions are:
x1 = (-b + sqrt(Delta))/2a
x2 = (-b - sqrt(Delta))/2a
In which:
Delta = b² - 4ac
The equation for this system if:
-x² + 9x + 8 = 3x + 8
x² - 6x = 0
x(x - 6) = 0
Then:
x = 0 -> y = 3 x 0 + 8 = 8.x - 6 = 0 -> x = 6, y = 3 x 6 + 8 = 26.Which means that the correct option is:
Yes, they intersect at the coordinates (0, 8) and (6, 26).
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Solve the inequality by the method of intervals
The solution of the inequality by method of interval is x = [5,∝) and x = [-2,0]∪[8,-∝).
What is the solution of the given inequality problems ?The first inequality is (x² - 25)/(x + 10) ≥ 0
⇒ x² - 25 ≥ 0
⇒ x² ≥ 25
∴ x ≥ 5
Thus the domain of the given inequality from method of interval is x = [5,∝).
The second inequality is (x + 2)(x² - 64)/(x² + 45) ≤ 0
⇒ (x + 2)(x² - 64)≤ 0
Either (x + 2)≤ 0 or (x² - 64)≤ 0
Either x ≤ -2 or x ≤ 8
Thus , the combined domain of the given inequality from method of interval is x = [-2,0]∪[8,-∝) .
Therefore, the solution of the inequality by method of interval is x = [5,∝) and x = [-2,0]∪[8,-∝).
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solve the equation 13Z/Z+1 =11-3i, Z£C where Z=X+yi X&Y£R?
Presumably you mean the equation
[tex]\dfrac{13z}{z+1} = 11-3i[/tex]
Observe that for [tex]z\neq-1[/tex],
[tex]\dfrac{13z}{z+1} = \dfrac{13(z+1) - 13}{z+1} = 13 - \dfrac{13}{z+1}[/tex]
so we can simplify the equation to
[tex]13 - \dfrac{13}{z+1} = 11 - 3i \implies \dfrac{13}{z+1} = 13 - (11 - 3i) = 2+3i[/tex]
Multiply both sides by [tex]\frac{z+1}{2+3i}[/tex].
[tex]\dfrac{13}{z+1}=2+3i \implies \dfrac{13}{2+3i} = z+1 \implies z = \dfrac{13}{2+3i} - 1[/tex]
Rationalize the denominator by introducing its complex conjugate.
[tex]z = \dfrac{13(2-3i)}{(2+3i)(2-3i)} - 1 = \dfrac{26-39i}{2^2+3^2} - 1 = \boxed{1 - 3i}[/tex]
50 POINTS!!!!!!!!!!!!!!!!!!!!!!!! SOLVE THIS QUICKLY WITH AN EXPLANATION
Answer:
x = 30 degrees
Step-by-step explanation:
angles MNO, LNM, and KNL make a straight line: 105+45+y = 180 --> y = 30 degrees
KNL = 30 degrees
angle KLN is 90 degrees, a triangle is equal to 180 degrees
2x+30+90 = 180
2x = 60
x = 30 degrees
You have a credit card that charges an interest rate of 16.15% compounded monthly. The table below shows your activity for the month of April. Date Activity Amount Balance April 1 Beginning Balance 1,000.00 April 5 Purchase 29.10 1,029.10 April 12 Payment −225.00 804.10 April 15 Purchase 97.25 901.35 April 23 Purchase 19.00 920.35 April 26 Purchase 28.40 948.75 April 30 Ending Balance 948.75 What is the average daily balance for this account? $ What is the finance charge for the month of April? $
1. The average daily balance for this credit card account is $187.74.
2. The finance charge for the month of April is $249.22.
How is the average daily balance calculated?The average daily balance for April is the total of the balance for every day in the billing cycle divided by the number of days in the billing cycle.
A billing cycle for a credit card account is usually one month.
Data and Calculations:Interest rate = 16.15% compounded monthly
Date Activity Amount Balance
April 1 Beginning Balance $1,000.00
April 5 Purchase 29.10 1,029.10
April 12 Payment −225.00 804.10
April 15 Purchase 97.25 901.35
April 23 Purchase 19.00 920.35
April 26 Purchase 28.40 948.75
April 30 Ending Balance 948.75
Total $5,632.05
Average daily balance = $187.735 ($5,632.05/30 days)
Daily interest rate = 0.04425 (16.15%/365)
Finance charge = $249.22 ($187.74 x 0.04425 30 days)
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Solve the following system of equations.
d + e = 6
d – e = 4
infinite number of solutions
infinite number of solutions
(5, 1)
(5, 1)
no solution
no solution
(3, –1)
(3, –1)
Answer: (5,1)
Step-by-step explanation:
Adding the equations gives 2d=10, and thus d=5.
So, it follows e=1.
Therefore, the solution is (5,1).
The solution is (5,1)
Given that d+e =6 and d-e =4
We need to find the solution of the given equations
Elimination method is used to eliminate a variable in an equation to find the value of another variable
Now here ,
We we will solve the equation i.e
d+e = 6
d-e = 4
Solving this equation We get,
2d = 10
Therefore,
d= 5
Now substituting the value of d in equation d+e =6
therefore,
5+e =6
Therefore e = 1
Hence the solution is (5,1)
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Someone please help with this question!
The value of angle 20 is as follows:
m∠20 = 50.9 degrees.
How to find angles when parallel lines are cut by a transversal?When parallel lines are cut by a transversal, angle relationships are formed. This include corresponding angles, alternate angles , vertical angles etc.
Therefore, line l and m are parallel lines cut by the two transversal.
Hence,
m∠6 ≅ m∠20 (alternate angles)
Therefore, alternate angles are congruent
m∠20 = 2x - 4
m∠19 + m∠20 = 180 (angles on a straight line)
5x - 8 + 2x - 4 = 180
7x - 12 = 180
7x = 180 + 12
7x = 192
x = 192 / 7
x = 27.4285714286
x = 27.43
Therefore,
m∠20 = 2(27.43) - 4
m∠20 = 50.8571428571
m∠20 = 50.9 degrees.
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What is the inverse of the function f (x) = 3(x + 5)2 – 4, such that x ≤ –5?
inverse of f of x is equal to negative 5 plus the square root of the quantity x plus 4 all over 3 end quantity
inverse of f of x is equal to negative 5 minus the square root of the quantity x plus 4 all over 3 end quantity
inverse of f of x is equal to negative 5 plus the square root of the quantity x over 3 plus 4 end quantity
inverse of f of x is equal to negative 5 minus the square root of the quantity x over 3 plus 4 end quantity
The equation of the inverse function of the function f(x) = 3(x + 5)^2 - 4 is f-1(x) = -5 + √[1/3(x + 4)]
What are inverse functions?Inverse functions are the opposite of an original equation. This means that for a function f(x), the inverse of the function f(x) is f-(x); it also represents the opposite function
How to determine the inverse functions?The function f(x) is given as
f(x) = 3(x + 5)^2 - 4
Express f(x) as y
So, we have
y = 3(x + 5)^2 - 4
Swap the positions of x and y
So, we have
x = 3(y + 5)^2 - 4
Add 4 to both sides of the equation
So, we have
x + 4 = 3(y + 5)^2
Divide through by 3
So, we have
1/3(x + 4) = (y + 5)^2
Take the square root of both sides
So, we have
√[1/3(x + 4)] = y + 5
Subtract 5 from both sides of the equation
So, we have
y = -5 + √[1/3(x + 4)]
Rewrite as
f-1(x) = -5 + √[1/3(x + 4)]
Hence, the equation of the inverse function of the function f(x) = 3(x + 5)^2 - 4 is f-1(x) = -5 + √[1/3(x + 4)]
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Which of the following functions has an initial value of -1/2 and a rate of change of -2?
The ratio of the weight of an object
The weighing of the man on the earth given the ratio is 174 pounds
Ratioratio of the weight of an object on the earth to the moon = 6:1Weight of a man on the moon = 29 poundsWeight of the man on Earth = xEquate the ratio
6 : 1 = x : 29
6/1 = x/29
cross product6 × 29 = 1 × x
174 = x
x = 174 pounds
Complete question:
The ratio of the weight of an object on the earth to the moon is 6:1. If a man weighs 29 pounds on the moon, calculate his weighing on the earth.
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A line is parallel to the line y = x and passes
through the parabola y = x² - 3 at
(4, 13). What is the other point at which the two
functions intersect?
Answer:
Step-by-step explanation:
hello here is an solution
The sum
of three numbers is 24. If two numbers are 16 and 22, what is the third?
Answer:
-14
Step-by-step explanation:
[tex]16+22+x=24\\38+x=24\\x=-14[/tex]
The terminal side of Ø is in quadrant II and cos ø = -5/13 What is sin ø?
the value of sin ∅ is 12/ 13
Quadrants and the "cast" Rule:In the first quadrant, the values for sin, cos, and tan are positive.In the second quadrant, the values for sin are positive only.In the third quadrant, the values for tan are positive only.In the fourth quadrant, the values for cos are positive only.From the given question,
We have, cos ∅= 5/13
From the trigonometric identities, we have that
[tex]sin^2\alpha + cos^2\alpha = 1[/tex]
Then , let's substitute the value of cos ∅
[tex]sin^2\alpha +(\frac{-5}{13} )^2 = 1[/tex]
Let make sin the subject of formula and find the squares of the fraction
sin²∅ = [tex]1 - \frac{25}{169}[/tex]
Find the LCM
sin²∅ = [tex]\frac{169 - 25}{169}[/tex]
Find the difference
sin²∅ = [tex]\frac{144}{169}[/tex]
Find the square root
sin∅ = [tex]\sqrt{\frac{144}{159} }[/tex]
sin∅ = [tex]\frac{12}{13}[/tex]
In quadrant II , sin is positive, so we have
sin ∅ = 12/ 13
Thus, the value of sin ∅ is 12/ 13
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