💡Okay, so I have this problem where α is an angle in the third quadrant, and I need to simplify the function f(α) which is given as a ratio of some trigonometric functions. Then, using a specific value for cos(α - 3π/2), I have to find the value of f(α). Hmm, let me try to break this down step by step.First, let me write down the function f(α):f(α) = [sin(α - π/2) * cos(3π/2 + α) * tan(π - α)] / [tan(-α - π) * sin(-π - α)]Alright, so I need to simplify this expression. I remember that there are some trigonometric identities that can help simplify these expressions. Let me recall them.Starting with the numerator:1. sin(α - π/2): I think there's an identity for sin(A - B). It's sin A cos B - cos A sin B. So, applying that here:sin(α - π/2) = sin α cos(π/2) - cos α sin(π/2)But cos(π/2) is 0 and sin(π/2) is 1, so this simplifies to:sin(α - π/2) = -cos αOkay, that's simpler.2. cos(3π/2 + α): Similarly, I can use the identity for cos(A + B) which is cos A cos B - sin A sin B.cos(3π/2 + α) = cos(3π/2) cos α - sin(3π/2) sin αI know that cos(3π/2) is 0 and sin(3π/2) is -1, so this becomes:cos(3π/2 + α) = 0 * cos α - (-1) * sin α = sin αWait, that seems right? Let me double-check. Cos(3π/2) is indeed 0, and sin(3π/2) is -1, so yeah, it becomes sin α. Okay.3. tan(π - α): There's an identity for tan(π - α). I think it's -tan α because tan is periodic with period π and tan(π - α) = -tan α.So, tan(π - α) = -tan αAlright, so putting it all together, the numerator becomes:sin(α - π/2) * cos(3π/2 + α) * tan(π - α) = (-cos α) * sin α * (-tan α)Multiplying these together:(-cos α) * sin α = -cos α sin αThen, multiplying by (-tan α):(-cos α sin α) * (-tan α) = cos α sin α tan αHmm, okay, so the numerator simplifies to cos α sin α tan α.Now, let's look at the denominator:1. tan(-α - π): Let's see, tan is periodic with period π, so tan(-α - π) = tan(-α). Also, tan is an odd function, so tan(-α) = -tan α. Therefore, tan(-α - π) = tan(-α) = -tan α.2. sin(-π - α): Similarly, sin is an odd function, so sin(-π - α) = -sin(π + α). Now, sin(π + α) is -sin α because sine is negative in the third quadrant. So, sin(-π - α) = -(-sin α) = sin α.Wait, let me make sure. sin(-π - α) = sin(-(π + α)) = -sin(π + α). And sin(π + α) = -sin α, so overall, sin(-π - α) = -(-sin α) = sin α. Yeah, that seems right.So, the denominator is:tan(-α - π) * sin(-π - α) = (-tan α) * sin α = -tan α sin αPutting it all together, the function f(α) becomes:Numerator: cos α sin α tan αDenominator: -tan α sin αSo, f(α) = [cos α sin α tan α] / [-tan α sin α]I can see that sin α and tan α appear in both numerator and denominator, so they can be canceled out, right?So, canceling sin α and tan α:f(α) = cos α / (-1) = -cos αWait, so f(α) simplifies to -cos α? That seems too simple. Let me check my steps again.Starting with the numerator:sin(α - π/2) = -cos αcos(3π/2 + α) = sin αtan(π - α) = -tan αSo, numerator: (-cos α) * sin α * (-tan α) = (-cos α)(sin α)(-tan α) = cos α sin α tan αDenominator:tan(-α - π) = tan(-α) = -tan αsin(-π - α) = sin αSo, denominator: (-tan α)(sin α) = -tan α sin αThus, f(α) = [cos α sin α tan α] / [-tan α sin α] = [cos α] / [-1] = -cos αYes, that seems correct. So, f(α) simplifies to -cos α.Alright, that was part (1). Now, part (2) says: If cos(α - 3π/2) = 1/5, find the value of f(α).Hmm, so we have cos(α - 3π/2) = 1/5. I need to find f(α) which is -cos α, so I need to find cos α first.Let me recall that cos(θ - 3π/2) can be rewritten using the cosine of a difference identity.cos(θ - 3π/2) = cos θ cos(3π/2) + sin θ sin(3π/2)But cos(3π/2) is 0 and sin(3π/2) is -1, so this simplifies to:cos(θ - 3π/2) = 0 + sin θ*(-1) = -sin θSo, cos(α - 3π/2) = -sin α = 1/5Therefore, -sin α = 1/5 => sin α = -1/5Okay, so sin α is -1/5. Since α is in the third quadrant, both sine and cosine are negative, which matches sin α being negative.Now, I need to find cos α. I can use the Pythagorean identity:sin² α + cos² α = 1We know sin α = -1/5, so sin² α = (1/5)² = 1/25Therefore, cos² α = 1 - sin² α = 1 - 1/25 = 24/25So, cos α = ±√(24/25) = ±(2√6)/5But since α is in the third quadrant, cosine is negative, so cos α = -2√6/5Therefore, f(α) = -cos α = -(-2√6/5) = 2√6/5So, f(α) is 2√6/5.Wait, let me just recap to make sure I didn't make any mistakes.1. Simplified f(α) to -cos α.2. Given cos(α - 3π/2) = 1/5, which translates to -sin α = 1/5, so sin α = -1/5.3. Using Pythagorean identity, found cos α = -2√6/5 because α is in the third quadrant.4. Then, f(α) = -cos α = 2√6/5.Yes, that seems consistent. I don't see any errors in my reasoning.**Final Answer**(1) boxed{-cos(alpha)}(2) boxed{dfrac{2sqrt{6}}{5}}

💡First, I observe that the given expression ((x^4 + 50x^2 + 625)(x^2 - 25)) resembles the form of the difference of cubes formula, which is ((a - b)(a^2 + ab + b^2) = a^3 - b^3). To apply this formula, I identify (a) and (b) in the given expression.I set (a = x^2) and (b = 25). Substituting these values into the difference of cubes formula, I get:[(x^2 - 25)(x^4 + 50x^2 + 625) = (x^2)^3 - 25^3]Next, I calculate each term:[(x^2)^3 = x^6][25^3 = 15625]Finally, I subtract the second term from the first to obtain the simplified expression:[x^6 - 15625]

💡Alright, so I've got this problem here about an even function, f(x), which is defined for all real numbers. It says that f(x) is increasing on the interval (-∞, 0]. I need to figure out the relationship between three values: a, which is f(1/2); b, which is f(1); and c, which is f(-3). The options are A: c < b < a; B: b < c < a; C: b < a < c; D: a < b < c.Okay, let's break this down step by step. First, I remember that an even function satisfies the condition f(x) = f(-x) for all x in its domain. So, that means f(-3) is the same as f(3). That's helpful because it allows me to express all the function values in terms of positive arguments, which might make it easier to compare them.Now, the function is increasing on (-∞, 0]. Since it's increasing on that interval, that means as x increases towards 0, f(x) also increases. But since it's an even function, what does that imply about its behavior on the interval [0, ∞)? Well, if f(x) is increasing as x approaches 0 from the negative side, then on the positive side, as x increases away from 0, f(x) should decrease. So, f(x) is decreasing on [0, ∞). That makes sense because even functions are symmetric about the y-axis, so the behavior on the right side of the y-axis is a mirror image of the left side.So, to summarize: f(x) is increasing on (-∞, 0] and decreasing on [0, ∞). Therefore, as x increases from 0 to +∞, f(x) decreases.Given that, let's look at the values we need to compare: a = f(1/2), b = f(1), and c = f(-3) = f(3). So, we're comparing f(1/2), f(1), and f(3).Since f(x) is decreasing on [0, ∞), the larger the x-value, the smaller the function value. So, let's order the x-values: 1/2 is less than 1, which is less than 3. Therefore, f(1/2) should be greater than f(1), which in turn should be greater than f(3). So, putting that together, we have f(3) < f(1) < f(1/2), which translates to c < b < a.Wait, let me make sure I'm not mixing up anything here. So, f(x) is decreasing on [0, ∞), meaning that as x increases, f(x) decreases. So, the smaller the x, the larger the f(x). Therefore, since 1/2 < 1 < 3, it follows that f(1/2) > f(1) > f(3). So, yes, c < b < a.Let me double-check if I've considered all the properties correctly. The function is even, so f(-3) = f(3). It's increasing on (-∞, 0], which implies it's decreasing on [0, ∞). So, as x increases from 0 to +∞, f(x) decreases. Therefore, the larger the positive x, the smaller the f(x). So, 1/2 is smaller than 1, which is smaller than 3, so f(1/2) is larger than f(1), which is larger than f(3). So, c < b < a, which is option A.I think that makes sense. I don't see any mistakes in my reasoning. So, I feel confident that the correct answer is A: c < b < a.

💡First, I need to calculate the interest the person pays for borrowing Rs. 4000 at 4% per annum for 2 years using the simple interest formula.Next, I will calculate the interest the person earns by lending the same amount at 6% per annum for 2 years using the same simple interest formula.After finding both the interest paid and earned, I will determine the total gain over the 2-year period by subtracting the interest paid from the interest earned.Finally, to find the annual gain, I will divide the total gain by 2.

💡Alright, so I've got this problem here where I need to find the value of ( b^{2014} - a^{2013} ) given that two sets ( M ) and ( N ) are equal. Let me try to unpack this step by step.First, let's write down what we know:- ( M = {1, a + b, a} )- ( N = {0, frac{b}{a}, b} )- ( M = N )- ( a ) and ( b ) are real numbers.Since ( M ) and ( N ) are equal sets, they must contain exactly the same elements, although not necessarily in the same order. That means every element in ( M ) must be in ( N ) and vice versa.Looking at ( M ), it has three elements: 1, ( a + b ), and ( a ). On the other hand, ( N ) has three elements: 0, ( frac{b}{a} ), and ( b ).So, for these sets to be equal, each element in ( M ) must match an element in ( N ). Let's try to figure out which elements correspond to each other.First, notice that ( M ) contains the number 1, while ( N ) contains 0. Since 1 is not equal to 0, 1 must correspond to one of the other elements in ( N ), which are ( frac{b}{a} ) or ( b ). Similarly, ( N ) contains 0, which must correspond to one of the elements in ( M ), which are 1, ( a + b ), or ( a ).So, let's consider the possibilities:1. **Case 1: ( 1 = frac{b}{a} )** If ( 1 = frac{b}{a} ), then ( b = a ). Now, let's see what else we can figure out. Since ( N ) has 0, which must be equal to one of the elements in ( M ). The elements in ( M ) are 1, ( a + b ), and ( a ). We already have ( b = a ), so ( a + b = a + a = 2a ). So, ( 2a ) must be equal to 0 because that's the only way 0 can be in ( M ). Therefore, ( 2a = 0 ) implies ( a = 0 ). Wait, but if ( a = 0 ), then ( frac{b}{a} ) would be undefined because we can't divide by zero. That's a problem because ( N ) is supposed to be a set of real numbers, and division by zero isn't allowed. So, this case leads to a contradiction. Therefore, ( 1 ) cannot be equal to ( frac{b}{a} ).2. **Case 2: ( 1 = b )** If ( 1 = b ), then ( b = 1 ). Now, let's see what else we can figure out. Again, ( N ) has 0, which must be equal to one of the elements in ( M ). The elements in ( M ) are 1, ( a + b ), and ( a ). Since ( b = 1 ), ( a + b = a + 1 ). So, we have two possibilities for where 0 comes from in ( M ): - **Subcase 2a: ( a + 1 = 0 )** If ( a + 1 = 0 ), then ( a = -1 ). Now, let's check the remaining elements. In ( M ), we have 1, ( a + b = 0 ), and ( a = -1 ). So, ( M = {1, 0, -1} ). In ( N ), we have 0, ( frac{b}{a} = frac{1}{-1} = -1 ), and ( b = 1 ). So, ( N = {0, -1, 1} ). Comparing ( M ) and ( N ), they are indeed equal: both contain 0, 1, and -1. So, this seems to work. Let's just double-check that all elements match: - 1 is in both sets. - 0 is in both sets. - -1 is in both sets. Perfect, so this case works without any contradictions. - **Subcase 2b: ( a = 0 )** If ( a = 0 ), then ( frac{b}{a} ) is undefined, which is not allowed because ( N ) must be a set of real numbers. So, this subcase is invalid. Therefore, the only valid subcase is Subcase 2a, where ( a = -1 ) and ( b = 1 ).3. **Case 3: ( 1 = a )** Wait, hold on. I initially considered ( 1 ) being equal to either ( frac{b}{a} ) or ( b ), but I didn't consider ( 1 ) being equal to ( a ). Let me check that as well. If ( 1 = a ), then ( a = 1 ). Now, ( M ) becomes ( {1, 1 + b, 1} ). But sets cannot have duplicate elements, so ( M = {1, 1 + b} ). However, ( N ) has three elements: 0, ( frac{b}{1} = b ), and ( b ). So, ( N = {0, b} ). Wait, but ( M ) has two elements and ( N ) has two elements, but originally, ( M ) was supposed to have three elements. This suggests that ( 1 + b ) must be equal to either 1 or ( a ), but since ( a = 1 ), ( 1 + b ) must be equal to 1 or 1. So, ( 1 + b = 1 ) implies ( b = 0 ). Let's see what happens if ( b = 0 ). Then, ( N = {0, 0, 0} ), but sets cannot have duplicate elements, so ( N = {0} ). Meanwhile, ( M = {1, 1, 1} = {1} ). So, ( M ) and ( N ) would both be single-element sets, but ( M = {1} ) and ( N = {0} ), which are not equal. Therefore, this case doesn't work either.So, after considering all possible cases, the only valid solution is when ( a = -1 ) and ( b = 1 ).Now, let's compute ( b^{2014} - a^{2013} ):- ( b = 1 ), so ( b^{2014} = 1^{2014} = 1 ).- ( a = -1 ), so ( a^{2013} = (-1)^{2013} = -1 ) because any negative number raised to an odd power remains negative.Therefore, ( b^{2014} - a^{2013} = 1 - (-1) = 1 + 1 = 2 ).So, the final answer is 2.**Final Answer**boxed{2}

💡Alright, so I have this problem about a hexagon inscribed in a circle. The hexagon has alternate sides of lengths 4 and 6. It's split into two quadrilaterals by a chord that passes through the intersection of the diagonals of the hexagon. One part has sides 4, 4, 4, and the other has sides 6, 6, 6. I need to find the length of this chord, express it as a ratio m/n where m and n are coprime positive integers, and then find m + n.First, let me visualize the hexagon. It's inscribed in a circle, so all its vertices lie on a single circle. The sides alternate between 4 and 6. Let me label the vertices as A, B, C, D, E, F in a clockwise direction. So, sides AB, BC, CD are 4 units each, and sides DE, EF, FA are 6 units each.Now, the hexagon is split into two quadrilaterals by a chord that passes through the intersection of the diagonals. The intersection of the diagonals... Hmm, in a hexagon, the diagonals can intersect at various points, but in this case, since it's inscribed in a circle, maybe it's referring to the intersection of two specific diagonals.Wait, the problem says the chord passes through the intersection of the diagonals. So, perhaps the diagonals intersect at a point inside the circle, and the chord passes through that point. This chord divides the hexagon into two quadrilaterals: one with sides 4, 4, 4 and another with sides 6, 6, 6.Let me think about the structure. If the chord passes through the intersection of the diagonals, maybe it's connecting two non-adjacent vertices? Or perhaps it's a chord that isn't a side or a diagonal but still passes through the intersection point.Wait, the hexagon is split into two quadrilaterals. So, each quadrilateral must have four sides. One quadrilateral has three sides of length 4 and one side which is the chord, and the other quadrilateral has three sides of length 6 and the same chord.Wait, actually, the problem says the hexagon is split into two quadrilaterals by a chord, which divides it into one part with sides 4, 4, 4 and another part with sides 6, 6, 6. So, each quadrilateral has three sides of the original hexagon and the chord as the fourth side.So, one quadrilateral has sides 4, 4, 4, and the chord, and the other has sides 6, 6, 6, and the chord. Therefore, the chord is the fourth side for both quadrilaterals.So, the chord is connecting two vertices such that on one side, it's part of three sides of length 4, and on the other side, it's part of three sides of length 6.Let me try to label the hexagon again. Let me assume that the chord connects vertex A to vertex D. So, chord AD. Then, quadrilateral AB CD would have sides AB=4, BC=4, CD=4, and AD. The other quadrilateral would be DEFA, with sides DE=6, EF=6, FA=6, and AD.Wait, but in that case, chord AD is the same for both quadrilaterals. So, if I can find the length of AD, that would be the chord.But is that the case? Or is the chord a different one?Wait, the problem says the chord passes through the intersection of the diagonals. So, maybe it's not just a single diagonal but a chord that passes through where two diagonals intersect.In a hexagon, the diagonals can intersect at various points. For example, in a regular hexagon, the diagonals intersect at the center, but this hexagon isn't regular since the sides alternate between 4 and 6.So, perhaps the diagonals intersect at some point inside the circle, and the chord passes through that intersection point.Wait, maybe it's similar to the regular hexagon where the diagonals intersect at the center, but in this case, it's not regular, so the intersection point might not be the center.But maybe it's still symmetric in some way.Alternatively, perhaps the chord is the line connecting the midpoints of the arcs subtended by the sides of length 4 and 6.Wait, maybe I should consider the arcs corresponding to the sides.Since the hexagon is inscribed in a circle, each side corresponds to an arc. The length of the side is related to the length of the arc it subtends. The longer the side, the larger the arc it subtends.Given that the sides alternate between 4 and 6, the arcs subtended by sides of length 4 will be smaller than those subtended by sides of length 6.Let me denote the central angles corresponding to sides of length 4 as θ and those corresponding to sides of length 6 as φ.Since it's a hexagon, there are six sides, so the sum of all central angles is 360 degrees.Given that the sides alternate, we have three sides of length 4 and three sides of length 6. Therefore, the total central angles would be 3θ + 3φ = 360 degrees, so θ + φ = 120 degrees.So, each pair of consecutive sides (4 and 6) subtends a total angle of 120 degrees at the center.Now, the chord that splits the hexagon into two quadrilaterals with sides 4,4,4 and 6,6,6 must be connecting two vertices such that on one side, it's connected to three sides of length 4, and on the other side, three sides of length 6.So, if I connect vertex A to vertex D, then quadrilateral ABCD has sides AB=4, BC=4, CD=4, and AD. The other quadrilateral DEFA has sides DE=6, EF=6, FA=6, and AD.Therefore, chord AD is the one that splits the hexagon into two quadrilaterals with sides 4,4,4 and 6,6,6.Therefore, I need to find the length of chord AD.Since the hexagon is inscribed in a circle, chord AD subtends an arc from A to D. Let me find the measure of that arc.From vertex A to D, there are three sides in between: AB, BC, CD. Each of these sides subtends angles θ, φ, θ respectively.Wait, no. Wait, sides AB, BC, CD are 4,4,4, so each subtends angle θ. Therefore, the arc from A to D is 3θ.Similarly, the arc from D back to A would be 3φ.But since θ + φ = 120 degrees, as established earlier, the total circumference is 360 degrees.Therefore, arc AD is 3θ, and arc DA is 3φ.But 3θ + 3φ = 360 degrees, so θ + φ = 120 degrees, which is consistent.Therefore, chord AD subtends an arc of 3θ.But I need to find the length of chord AD. The length of a chord is given by 2R sin(α/2), where α is the central angle subtended by the chord, and R is the radius of the circle.So, chord AD = 2R sin(3θ/2).But I don't know R or θ yet.Similarly, sides of length 4 and 6 correspond to central angles θ and φ, respectively.The length of a side is given by 2R sin(α/2), where α is the central angle.Therefore, for side AB=4: 4 = 2R sin(θ/2) => sin(θ/2) = 2/R.Similarly, for side DE=6: 6 = 2R sin(φ/2) => sin(φ/2) = 3/R.Also, we know that θ + φ = 120 degrees.So, we have two equations:1. sin(θ/2) = 2/R2. sin(φ/2) = 3/RAnd θ + φ = 120 degrees.Let me denote θ = 2α, so θ/2 = α, and φ = 120 - θ = 120 - 2α.Therefore, φ/2 = 60 - α.So, equation 1 becomes sin(α) = 2/REquation 2 becomes sin(60 - α) = 3/RSo, we have:sin(60 - α) = 3/RBut sin(60 - α) can be expanded using sine subtraction formula:sin(60 - α) = sin60 cosα - cos60 sinαWe know sin60 = √3/2, cos60 = 1/2.So,sin(60 - α) = (√3/2)cosα - (1/2)sinαBut from equation 1, sinα = 2/R, so let's denote sinα = 2/R.Let me denote sinα = s = 2/R, so cosα = sqrt(1 - s²) = sqrt(1 - 4/R²).Similarly, sin(60 - α) = 3/R.So,(√3/2)cosα - (1/2)sinα = 3/RSubstituting sinα = 2/R and cosα = sqrt(1 - 4/R²):(√3/2)sqrt(1 - 4/R²) - (1/2)(2/R) = 3/RSimplify:(√3/2)sqrt(1 - 4/R²) - (1/R) = 3/RBring the (1/R) to the right side:(√3/2)sqrt(1 - 4/R²) = 3/R + 1/R = 4/RSo,(√3/2)sqrt(1 - 4/R²) = 4/RMultiply both sides by 2/√3:sqrt(1 - 4/R²) = (8)/(R√3)Square both sides:1 - 4/R² = 64/(3R²)Multiply both sides by 3R² to eliminate denominators:3R² - 12 = 64So,3R² = 64 + 12 = 76Therefore,R² = 76/3So,R = sqrt(76/3) = (2*sqrt(57))/3Wait, let me check that:76 divided by 3 is approximately 25.333, so sqrt(76/3) is sqrt(25.333) which is approximately 5.033.But let me compute sqrt(76/3):76 = 4*19, so sqrt(76/3) = sqrt(4*19/3) = 2*sqrt(19/3) = 2*sqrt(57)/3.Wait, sqrt(19/3) is sqrt(57)/3, because 19/3 = (19*3)/(3*3) = 57/9, so sqrt(57)/3.Yes, so R = 2*sqrt(57)/3.Now, going back, we need to find chord AD, which subtends an arc of 3θ.But θ = 2α, so 3θ = 6α.So, chord AD = 2R sin(3θ/2) = 2R sin(3α).Wait, chord length formula is 2R sin(α/2), where α is the central angle.So, chord AD subtends a central angle of 3θ.Wait, no, chord AD subtends an arc from A to D, which is 3θ, as each side AB, BC, CD subtends θ.So, central angle for chord AD is 3θ.Therefore, chord AD = 2R sin(3θ/2).But θ = 2α, so 3θ/2 = 3α.Therefore, chord AD = 2R sin(3α).We know R = 2*sqrt(57)/3, so let's compute sin(3α).We have sinα = 2/R = 2/(2*sqrt(57)/3) = 3/sqrt(57) = sqrt(57)/19 * 3 = 3*sqrt(57)/57 = sqrt(57)/19.Wait, let me compute sinα:sinα = 2/R = 2/(2*sqrt(57)/3) = (2)*(3)/(2*sqrt(57)) = 3/sqrt(57) = sqrt(57)/19 * 3.Wait, 3/sqrt(57) can be rationalized as 3*sqrt(57)/57 = sqrt(57)/19.Yes, so sinα = sqrt(57)/19.Therefore, cosα = sqrt(1 - sin²α) = sqrt(1 - 57/361) = sqrt((361 - 57)/361) = sqrt(304/361) = sqrt(304)/19.Simplify sqrt(304): 304 = 16*19, so sqrt(304) = 4*sqrt(19).Therefore, cosα = 4*sqrt(19)/19.Now, sin(3α) can be computed using the sine addition formula:sin(3α) = 3 sinα - 4 sin³α.So,sin(3α) = 3*(sqrt(57)/19) - 4*(sqrt(57)/19)^3Compute each term:First term: 3*(sqrt(57)/19) = 3 sqrt(57)/19Second term: 4*(sqrt(57)/19)^3 = 4*(57 sqrt(57))/(19^3) = 4*(57 sqrt(57))/(6859)Simplify 57/6859: 57 divides into 6859: 6859 /57 = 120.333... Wait, 57*117 = 6669, 57*118=6726, 57*119=6783, 57*120=6840, 57*121=6897. So, 57*119=6783, which is less than 6859. 6859 - 6783=76. 76 is 57 + 19. So, 6859=57*119 +76=57*119 +57 +19=57*120 +19. So, 57 doesn't divide 6859 evenly. Hmm, maybe I made a mistake.Wait, 19^3 is 6859, right? 19*19=361, 361*19=6859.So, 4*(sqrt(57)/19)^3 = 4*(57 sqrt(57))/(19^3) = 4*57 sqrt(57)/6859.Simplify 4*57=228.So, 228 sqrt(57)/6859.Now, let's see if 228 and 6859 have a common factor.6859 divided by 19 is 361, which is 19². 228 divided by 19 is 12. So, 228=19*12, 6859=19³.Therefore, 228/6859 = (19*12)/(19³) = 12/(19²) = 12/361.Therefore, 228 sqrt(57)/6859 = (12/361) sqrt(57).So, sin(3α) = 3 sqrt(57)/19 - 12 sqrt(57)/361.Convert 3 sqrt(57)/19 to 117 sqrt(57)/361 (since 3*19=57, 57*6=342, wait, no, 3/19 = 57/361? Wait, 3*19=57, so 3/19 = 57/361.Wait, 3 sqrt(57)/19 = (3*19 sqrt(57))/(19*19) = 57 sqrt(57)/361.Wait, no, that's not correct. Wait, 3 sqrt(57)/19 is equal to (3/19) sqrt(57). To express it over 361, multiply numerator and denominator by 19:(3*19 sqrt(57))/(19*19) = 57 sqrt(57)/361.Similarly, 12 sqrt(57)/361 is already over 361.Therefore, sin(3α) = 57 sqrt(57)/361 - 12 sqrt(57)/361 = (57 sqrt(57) - 12 sqrt(57))/361 = (45 sqrt(57))/361.So, sin(3α) = 45 sqrt(57)/361.Therefore, chord AD = 2R sin(3α) = 2*(2 sqrt(57)/3)*(45 sqrt(57)/361).Compute this:First, multiply constants:2*(2/3)*(45/361) = (4/3)*(45/361) = (180)/1083.Simplify 180/1083: divide numerator and denominator by 3: 60/361.Now, multiply sqrt(57)*sqrt(57) = 57.So, chord AD = (60/361)*57.Compute 60*57: 60*50=3000, 60*7=420, total=3420.Therefore, chord AD = 3420/361.Simplify 3420/361: divide numerator and denominator by GCD(3420,361). Let's find GCD(3420,361).361 divides into 3420 how many times? 361*9=3249, 3420-3249=171.Now, GCD(361,171). 361 divided by 171 is 2 with remainder 19 (171*2=342, 361-342=19).GCD(171,19). 171 divided by 19 is 9 with remainder 0. So, GCD is 19.Therefore, divide numerator and denominator by 19:3420 ÷19= 180, 361 ÷19=19.So, chord AD=180/19.Therefore, the length of the chord is 180/19, which is already in lowest terms since 180 and 19 are coprime (19 is prime and doesn't divide 180).Thus, m=180, n=19, so m+n=199.Wait, but let me double-check my calculations because this seems a bit involved.First, R was found to be 2 sqrt(57)/3.Then, sinα = sqrt(57)/19, cosα=4 sqrt(19)/19.Then, sin(3α)=3 sinα -4 sin³α=3*(sqrt(57)/19) -4*(sqrt(57)/19)^3.Computed sin³α as (57 sqrt(57))/57³? Wait, no, sinα= sqrt(57)/19, so sin³α=(sqrt(57))^3 /19³=57 sqrt(57)/6859.Then, 4 sin³α=4*57 sqrt(57)/6859=228 sqrt(57)/6859=12 sqrt(57)/361.Then, 3 sinα=3 sqrt(57)/19=57 sqrt(57)/361.So, sin(3α)=57 sqrt(57)/361 -12 sqrt(57)/361=45 sqrt(57)/361.Then, chord AD=2R sin(3α)=2*(2 sqrt(57)/3)*(45 sqrt(57)/361)= (4 sqrt(57)/3)*(45 sqrt(57)/361).Multiply constants: 4*45=180, denominators:3*361=1083.Multiply sqrt(57)*sqrt(57)=57.So, chord AD=180*57 /1083.Compute 180*57=10260.10260/1083.Simplify: divide numerator and denominator by 3: 3420/361.As before, 3420 ÷19=180, 361 ÷19=19.So, 180/19.Yes, that seems correct.Therefore, the length of the chord is 180/19, so m=180, n=19, m+n=199.