A recursive definition for the set S can be given as follows:
What is recursion?
Recursion is a programming technique where a function calls itself to solve a problem.
Base case: (1, n) is in S for all positive integers n, since 1 is a factor of all positive integers.
Recursive case: If (a, b) is in S, then (a', b) is in S for all positive integers a' that are factors of a, and (a, b') is in S for all positive integers b' that are multiples of b.
In other words, the set S contains all pairs (a,b) where a is a positive integer that divides b, and b can be obtained by multiplying any such a with another positive integer. The base case includes all pairs where a=1 and b is any positive integer.
The recursive case states that if (a,b) is in S, then all pairs where a' is a factor of a and b is a positive integer such that b=a'b are also in S, as well as all pairs where b' is a multiple of b and a is a positive integer that divides b'.
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answer and explain in detail
Answer:
The area of right-angled triangle is 30 cm².
Step-by-step explanation:
Given : Here we given that the base and hypotenuse of triangle 12 cm and 13 cm respectively.To Find : We have to find the area of right-angled triangle but before we will find the height of the triangle.Solution :By using Pythagoras Theorem we will find the hypotenuse of triangle :
[tex] \sf{\longrightarrow{{(Hypotenuse)}^{2} = {(Height)}^{2} + {(Base)}^{2}}}[/tex]
Substituting all the given values in the formula to find hypotenuse :
Hypotenuse = 13 cmBase = 12 cm[tex]\sf{\longrightarrow{{(Hypotenuse)}^{2} = {(Height)}^{2} + {(Base)}^{2}}}[/tex]
[tex]\sf{\longrightarrow{{(13)}^{2} = {(Height)}^{2} + {(12)}^{2}}}[/tex]
[tex]\sf{\longrightarrow{(13 \times 13) = {(Height)}^{2} + {(12 \times 12)}}}[/tex]
[tex]\sf{\longrightarrow{(169) = {(Height)}^{2} + {(144)}}}[/tex]
[tex]\sf{\longrightarrow{{(Height)}^{2} = 169 - 144}}[/tex]
[tex]\sf{\longrightarrow{{(Height)}^{2} = 25}}[/tex]
[tex]\sf{\longrightarrow{{(Height)} = \sqrt{25}}}[/tex]
[tex]\sf{\longrightarrow{\underline{\underline{\red{(Height) = 5 \: cm}}}}}[/tex]
Hence, the height of triangle is 5 cm.
[tex]\begin{gathered} \end{gathered}[/tex]
Now, calculating the area of right-angled triangle by substituting all the given values in the formula :
[tex]\dashrightarrow{\sf{Area_{(\triangle)} = \dfrac{1}{2}bh}}[/tex]
b (Base) = 12 cmh (Height) = 5 cm[tex]\dashrightarrow{\sf{Area_{(\triangle)} = \dfrac{1}{2}bh}}[/tex]
[tex]\dashrightarrow{\sf{Area_{(\triangle)} = \dfrac{1}{2} \times b \times h}}[/tex]
[tex]\dashrightarrow{\sf{Area_{(\triangle)} = \dfrac{1}{2} \times 12\times 5}}[/tex]
[tex]\dashrightarrow{\sf{Area_{(\triangle)} = 6\times 5}}[/tex]
[tex]\dashrightarrow{\sf{\underline{\underline{\pink{Area_{(\triangle)} =30 \: {cm}^{2}}}}}}[/tex]
Hence, the area of triangle is 30 cm².
—————————————————The price of an item has been reduced by 25%. The original price was $35
Please help, ASAP! This is for a College Algebra Class, No wrong answer.
The value of x in the equation is 2.10.
How to solve exponential equation?The exponential equation above can be solved as follows:
[tex]8^{\frac{4}{x} } = 53[/tex]
Therefore, let's log both sides
log [tex]8^{\frac{4}{x} }[/tex] = log 53
Hence,
4 / x log 8 = log 53
divide both sides by log 8
4 / x = log 53 / log 8
4 / x = 1.7242758696 / 0.90308998699
4 / x = 1.9092439208
cross multiply
1.902x = 4
x = 4 / 1.902
x = 2.09511837419
Therefore,
x = 2.10
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I am in need of this to be answered~
Transform this sentence into NNF:
~(~(P&~Q)v(R&~S))
Please answer, step by step to the answer.
Thank you!
~(~(P&~Q)v(R&~S)) transformation of this statment into NNF is ((~P v Q) & (~R v S))
How to transform the statement by using NNF?The transformations used to turn the given sentence into NNF:
Using the equivalence, remove the implication: p → q ≡ ~p ∨ q
≡ (((P&~Q)) & ~(R&~S))
Use De Morgan's law: ~(p ∨ q) ≡ ~p & ~q, and ~(p & q) ≡ ~p ∨ ~q
≡ ((~P v Q) & (~R v S))
Elimination by double negation:~~p ≡ p
≡ ((~P v Q) & (~R v S))
The resulting sentence, ~(~(P&~Q)v(R&~S)) is in NNF (Negation Normal Form), which limits negations to atomic propositions (P, Q, R, S) and logical connectives ¬, ∧, and ∨.
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Math: please answer this question very important for me, I’ll give brainliest if it’s correct!!
Q2. The average depth of the water in a port on a tidal river is 4 m. At low tide, the depth of the water is 2 m. One cycle is completed approximately every 12h.
(a) [4 marks] Find an equation of the depth, d(t) meters, with respect to the average depth, as a function of the time, t hours, after low tide, which occurred at 15: 00.
(b) [2 marks] Draw a graph of the function for 48 h after low tide.
A sports statistician determined that the probability of a certain rugby team winning its next match is 11/23. Find the odds against the team winning its next match.
The probability of the odds of the rugby team winning its next match is
12 to 11.
We have,
The probability of an event happening is defined as the ratio of the number of favorable outcomes to the total number of outcomes.
The odds against an event happening are defined as the ratio of the number of unfavorable outcomes to the number of favorable outcomes.
If the probability of the rugby team winning its next match is 11/23, then the probability of the team losing its next match is:
= 1 - 11/23
= 12/23.
The odds against the rugby team winning its next match are the ratio of the number of unfavorable outcomes (losing the match) to the number of favorable outcomes (winning the match), which is:
12/23 : 11/23
We can simplify this ratio by dividing both the numerator and denominator by 11:
(12/23) ÷ (11/23) : (11/23) ÷ (11/23)
Simplifying.
12/11 : 1/1
Therefore,
The probability of the odds of the rugby team winning its next match is
12 to 11.
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Solve the following system of linear equations 3x₁ + x₂ + 2x3= 6
4x₁ - 3x₂ + 5x3 = 8,
2x₁ + x₂ + 3x3 = 10
Okay, here are the steps to solve this system of linear equations:
3x1 + x2 + 2x3= 6
4x1 - 3x2 + 5x3 = 8,
2x1 + x2 + 3x3 = 10
1) Add 4 times the first equation to the second equation:
7x1 + 2x2 + 7x3 = 22
2) Add -2 times the first equation to the third equation:
5x1 - x2 + 5x3 = 12
3) Divide both sides by 5 to get:
x1 = 2
x2 = -2
x3 = 1
Therefore, the solution to the system is:
x1 = 2
x2 = -2
x3 = 1
Let me know if you have any other questions!
#9 75% of 1
#10 50% of 350 is
#11 What percent of 450 is 50
#9 75% of 1.
#10 50% of 350.
#11
To find out what percent of 450 is 50, we can set up a proportion
x/100 = 50/450
Simplifying the right-hand side of the equation by dividing both numerator and denominator by 50 gives
x/100 = 1/9
Multiplying both sides by 100 gives
x = 100/9
Hence, 50 is approximately 11.1% of 450.
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For a certain set of five numbers, the mean of all but the largest number is 80 and the
mean of all but the smallest number is 90. What is the range of the set of five numbers?
Thus, the range of the set of five numbers (a,b,c,d,e) is found as : 40.
Explain about the range of set:The difference in between highest and lowest numbers in a piece of data is known as the range. To locate it, sort the data first from least to greatest. Then take the difference between the least and largest number in the set.
The range is a straightforward assessment of the variation in values within a dataset. Simply take the difference between the two values, ignoring the others, to determine the range.
Let the 5 numbers be: (a,b,c,d,e)
a < b < c < d < e
Mean = sum of all observation / total number of observations
Equation for the mean except the largest number.
(a+b+c+d)/4 = 80
Now, Equation for the mean except the smallest number.
(b+c+d+e)/4 = 90
On simplifying each:
a + b + c + d = 320 ...eq 1
b + c + d + e = 360 ..eq 2
Range = max - min
Subtract eq 1 from eq 2
e - a = 40
Thus, the range of the set of five numbers (a,b,c,d,e) is found as : 40.
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Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).
The three (3) true statements about the quadratic function and its graph include the following:
A. The value of f(–10) = 82
B. The graph of the function is a parabola.
D. The graph contains the point (20, –8).
How to determine the true statements about this quadratic function?Generally speaking, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. For this quadratic function, the graph is a upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero.
Next, we would determine the other true statements about the graph of this quadratic function:
At point (-10, 82), we have:
Quadratic function, f(x) = 1/5 x² – 5x + 12
Quadratic function, f(x) = x²/5 – 5x + 12
Quadratic function, f(-10) = -10²/5 – 5(-10) + 12
Quadratic function, f(-10) = 82
At point (20, -8), we have the following:
Quadratic function, f(x) = 1/5 x² – 5x + 12
Quadratic function, f(x) = x²/5 – 5x + 12
Quadratic function, f(20) = 20²/5 – 5(20) + 12
Quadratic function, f(20) = -8
In conclusion, we can logically deduce that the graph of this quadratic function does not contain the point (0, 0) as shown in the image attached below.
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Is anyone here looking for a personal tutor?????
meAnswer: me
Step-by-step explanation: I need help
the temperature at noon was 10degree Celsius above 0 . if it decreases at the rate 2 degree Celsius per hour till midnight , at time the temperature would be 8 degree Celsius below zero ? what would be the temperature at mid-night?
Answer:
Time at which temperature will be -8 degrees Celsius = 21 00 (24 hour clock) or 9pm (12 hour clock)
Temperature at midnight = - 14°C
Step-by-step explanation:
The pic
Need to solve some questions can you Help me?
2nd Question
6 ÷ 2 ( 1 + 2 ) = ?
Answer:
Step-by-step explanation:
[tex]6/2 ( 1 + 2 ) = 3(3) = 9[/tex]
PLEASE SOLVE THIS....I WILL GIVE BRAINLIEST!!!
Answer:
Give me a sec
Step-by-step explanation:
The garment factory plans to make 690 sets of garments. It has been done for 5 days, with an average of 75 sets per day. The rest needs to be completed in 2.5 days, how many sets are done on average per day (step-by-step answer)
For the remaining set of garments, the average per day that they need to be completed would be = 126 sets/day.
How to calculate the average set made per day?The quantity of garments planned to be made by the factory = 690sets.
The rate of coverage for 5 days = 75sets/day.
That is for 5 days the number of garments covered = 75×5 = 375
The remaining sets = 690-375 = 315
The average per day to cover the remaining sets of garment = 315/2.5 = 126 sets/day
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The size P of a certain insect population at time t (in days) obeys the function P = 700e0.03t, After how many days will the population reach 1500?
the population will reach 1500 after about 30.1 days.
After how many days will the population reach 1500?We can solve for the time t by setting P = 1500 and solving for t:
1500 = 700[tex]e^{(0.03t)}[/tex]
Divide both sides by 700:
2.14 = [tex]e^{(0.03t)}[/tex]
Take the natural logarithm of both sides:
ln(2.14) = ln([tex]e^{(0.03t)}[/tex])
Using the logarithm property that ln([tex]e^{x}[/tex]) = x, we can simplify to:
ln(2.14) = 0.03t
Divide both sides by 0.03:
t = ln(2.14)/0.03
Using a calculator, we get:
t ≈ 30.1 days
Therefore, the population will reach 1500 after about 30.1 days.
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Winnie and Vanessa Co Ltd. borrowed an amount of money (P) from a rural bank in Miotso at i% p.a simply interest. The company paid A amount of money as well as the total amount of interest (I) to the bank at the end of n- years. Show that, p= A(1+in)-1
The amount of money borrowed by the company from the rural bank is:
P = [tex]A(1 + in)^{-1}[/tex]
What is Amount?
Amount is a term used to describe the total quantity or value of something. It can refer to a quantity of a physical substance or object, such as the amount of water in a bottle or the amount of money in a bank account.
Let P be the amount of money borrowed by Winnie and Vanessa Co Ltd from the rural bank at an interest rate of i% per annum.
The simple interest on this amount after n years will be:
I = P * i * n
The total amount to be paid by the company after n years will be:
A = P + I = P + P * i * n
We can simplify this expression by factoring out P:
A = P(1 + i * n)
Dividing both sides of this equation by (1 + i * n), we get:
P = A / (1 + i * n)
Therefore, the amount of money borrowed by the company from the rural bank is:
P = [tex]A(1 + in)^{-1}[/tex]
This is the desired result.
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Please help me
Decide the type of transformation and state the rule
The four translation rules are defined as follows:
Left a units: x -> x - a.Right a units: x -> x + a.Up a units: y -> y + a.Down a units: y -> y - a.Comparing the equivalent vertices, the translations are given as follows:
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A caterer made a sandwich 6' 6" long for a party. by request 3' 8" were added to the sandwich. what was the total length of the sandwich?
The total length of the sandwich, given the addition to the sandwich, would be 10. 17 feet.
How to find the length of the sandwich ?In order to accurately add lengths measured in feet and inches, it is necessary to transform the unit type prior to adding them together. As an example, take a sandwich that originally measured 6 feet 6 inches long with 3 feet 8 inches being added into it.
To combine these measurements, we must first convert inches to feet or vice versa for optimal results. Let's begin with transforming the inches to feet:
6 feet 6 inches = 6 + ( 6 / 12 ) feet = 6. 5 feet
3 feet 8 inches = 3 + ( 8 / 12 ) feet = 3. 67 feet
Then add the lengths to become :
6. 5 feet + 3. 67 feet = 10. 17 feet
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what is an equation of the image of the line after dilation
An equation of the image of the line after dilation is -30x - 20y = -8.
What is scale factor?In Mathematics and Geometry, a scale factor is the ratio of two corresponding side lengths or diameter in two similar geometric objects, which can be used to either vertically or horizontally enlarge (increase) or reduce (compress) a function representing their size.
Generally speaking, the transformation rule for the dilation of a geometric object based on a specific scale factor of 5 is given by this mathematical expression:
(x, y) → (kx, ky)
Where:
x and y represents the data points.k represents the scale factor.Therefore, the transformation rule for this dilation is given by;
(x, y) → (kx, ky)
(x, y) → (5(-6x), 5(-4y)) = -30x - 20y = -8.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
What is the unit sales of product A at the monthly break even point?
The unit sales of product A at the monthly break-even point is 193,750 units.
What is the unit sales of product A at the monthly break even point?To determine the unit sales of product A at the monthly break-even point, we need to calculate the total contribution margin that covers the monthly fixed costs of $387,500.
Given that the unit contribution margin for product A is $2 and the unit selling price for product A is $5, we can calculate the contribution margin ratio for product A as follows:
Contribution Margin Ratio for Product A:
= (Unit Contribution Margin for Product A) / (Unit Selling Price for Product A)
= $2 / $5
= 0.4
40%
Now, let's consider the sales ratios given in the problem statement. It states that the company sells two units of A for each unit of B, and three units of B for each unit of C. This means that the sales ratio for product A to product B is 2:1, and the sales ratio for product B to product C is 3:1.
Let's assume that the unit sales of product A at the monthly break-even point is "x". Then, the unit sales of product B would be "2x" (based on the sales ratio of 2:1 between A and B), and the unit sales of product C would be "6x" (based on the sales ratio of 3:1 between B and C).
The total contribution margin for product A at the monthly break-even point can be calculated as follows:
Total Contribution Margin for Product A = (Unit Contribution Margin for Product A) * (Unit Sales of Product A at Break-Even Point)
= $2 * x
Since the monthly fixed costs are $387,500, we can set up the equation for the break-even point as follows:
Total Contribution Margin for Product A = Monthly Fixed Costs
$2 * x = $387,500
Now we can solve for x, the unit sales of product A at the monthly break-even point:
$2 * x = $387,500
x = $387,500 / $2
x = 193,750
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For the rotation -757°, find the coterminal angle from 0° ≤ 0 < 360°, the
quadrant, and the reference angle.
To find the coterminal angle for a rotation of -757°, we can add or subtract multiples of 360° until we find an angle between 0° and 360° that is coterminal with -757°.
Adding 360° multiple times:
-757° + 360° = -397° (not coterminal)
-397° + 360° = -37° (not coterminal)
-37° + 360° = 323°
Therefore, the coterminal angle for -757° is 323°.
To find the quadrant in which 323° lies, we can note that 323° is between 270° and 360°, which means it lies in the fourth quadrant.
To find the reference angle for 323°, we can subtract 360° from 323° until we get an angle between 0° and 90°:
323° - 360° = -37° (not in the correct range)
-37° + 360° = 323°
So the reference angle for 323° is 37°.
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRoot x = –1 Plus or minus StartRoot StartFraction 4 Over 8 EndFraction EndRoot 8(x2 + 2x + 1) = 3 + 1 8(x2 + 2x) = –3
Answer:
Step-by-step explanation:
Patel is solving 8x2 + 16x + 3 = 0. Which steps could he use to solve the quadratic equation? Select three options. 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartR…
✓ Answer:8(x2 + 2x) = –3 8(x2 + 2x + 1) = –3 + 8 x = –1 Plus or minus StartRoot StartFraction 5 Over 8 EndFraction EndRootStep-by-step explanation:S…
Plot the numbers -1 1/2 and 2 3/4 on the number line below.
For the second number 2 3/4 you will find the 2 points, to the right from 0. Now count three small points to the left to get the 3/4 part
How to solveTo locate the value of -1 1/2 on the number line, first locate the point denoting -1 to the left of 0. Progressing to the second minor point on the right immediately after -1 will indicate this position.
For the second figure of 2 3/4, identify the point at 2 by moving to the right from 0. An additional count of three minor points towards the left reveals where the fraction of 3/4 is located. The exact spot is depicted here.
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The winner of a contest will be blindfolded and then allowed to reach into a hat containing 29 $1 bills and 7 $100 bills. The winner can remove 4 bills from the hat and keep the money. Find the probability that the winner will pull out at least 1 $100 bill. (See Example 5 in this section. Round your answer to the nearest tenth of a percent.)
The probability that the winner will pull out at least one $100 bill is 7/36, or 19.4%.
What is probability?The probability of an event occurring is determined by the number of favorable outcomes divided by the total number of possible outcomes.
In this problem, the favorable outcome is that the winner will pull out at least one $100 bill, while the total number of possible outcomes is the number of ways to select 4 bills from the hat, which is 36.
Thus, the probability of the winner pulling out at least one $100 bill is calculated by dividing the favorable outcome by the total number of possible outcomes.
Since there are 7 $100 bills in the hat, the favorable outcome is 7. The total number of possible outcomes is 36, which is calculated by using the formula nCr, where n is the total number of objects and r is the number of objects selected.
In this case, n = 36 and r = 4, so the equation is 36C4.
Thus, the probability that the winner will pull out at least one $100 bill is 7/36, or 19.4%.
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5. Shanequa graphed a triangle with the coordinates R(-2, 1), S(2, 1) and T(0, -3). She wanted to graph
a dilation of the figure with a scale factor of 2 centered at the origin so she graphed the points
R'(-2, 2), S'(2, 2) and T'(0, -6).
a. Is AR'S'T' a dilation of ARST? Why or why not?
b. If Shanequa was wrong, correct her answer by giving the correct coordinates for AR'S'T'.
c. Shanequa's teacher said Shanequa had made a common mistake. Explain her mistake.
a) No , it is not a dilation
b) The coordinates are R'(-2, 2), S'(2, 2), and T'(0, -6)
c) Shanequa's mistake is that she incorrectly applied the dilation transformation
Given data ,
a)
In ARST, the coordinates of the original triangle are given by R(-2, 1), S(2, 1), and T(0, -3), and in AR'S'T', the coordinates are R'(-2, 2), S'(2, 2), and T'(0, -6)
So , the dilated triangle is R'(-2, 2), S'(2, 2), and T'(0, -6)
b)
The coordinates for AR'S'T' can be calculated by multiplying the original coordinates of ARST by the scale factor of 2, centered at the origin.
R'(-4, 2), S'(4, 2), and T'(0, -6)
c)
The dilation transformation was applied wrongly by Shanequa. The coordinates of the original points in a dilation centred at the origin with a scale factor of two should be multiplied by the scale factor to obtain the coordinates of the dilated points.
Hence , the dilation is solved
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For questions 4 – 7, find the value of x.
The values of x in the isosceles trapezoids x = 32, x = 10.9 and x = 41
Calculating the values of x in the figuresAn isosceles trapezoid is a four-sided geometric shape that has two parallel sides and two non-parallel sides that are of equal length.
The parallel sides are called the bases of the trapezoid, while the non-parallel sides are called the legs and the angles in the same base are equal in measure
Using the above as a guide, we have the following:
2x + 14 = 78
x = 32
2 * (6x +32 + 4x + 39) = 360
x = 10.9
4x = 164
x = 41
The last details are not clear
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what is true about the modeling equation that best fits the data?
• It captures the overall trends and patterns in the data points. It does not necessarily pass through every single data point but approximates the shape and trajectory of the data.
• It has parameters that can be adjusted to optimize how well the model fits the data. Things like coefficients, exponents, intercepts, etc. The good parameter values will depend on the specific data set and model.
• It has an appropriate level of complexity for the data. It does not overfit or underfit the data. An overfit model captures every detail and noise, while an underfit model fails to capture the key dynamics.
• Key features like slope, curvature, periodicity, thresholds, etc. in the data are properly represented in the model.
• It provides insights or useful information, not just a mathematical abstraction of the data. A good model opens the black box and provides interpretable knowledge about the system or phenomenon that generated the data.
• Predictions or extrapolations from the model align well with any additional data points beyond the original data set. The model generalizes purposefully.
• There are objective metrics that can evaluate how good a particular model is for a data set, like R2, mean absolute error, Akaike information criterion, etc. The best model will optimize such metrics.
Given that sin theta = 15/17 with theta quadrant 1 , find cos 2theta
[tex]\textit{Double Angle Identities} \\\\ \cos(2\theta)= \begin{cases} \cos^2(\theta)-\sin^2(\theta)\\ 1-2\sin^2(\theta)&\leftarrow \textit{we'll use this one}\\ 2\cos^2(\theta)-1 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ 1-2\sin^2(\theta)\implies 1-2\left( \cfrac{15}{17} \right)^2\implies 1-2\left( \cfrac{225}{289} \right) \\\\\\ 1-\cfrac{450}{289}\implies \cfrac{289-450}{289}\implies -\cfrac{161}{289}[/tex]
I NEED HELP ASAP PLEASE HELP!!!!!!