💡Okay, so I have this problem where a point is given in rectangular coordinates as (5, 12), and I need to find the rectangular coordinates after transforming it using polar coordinates. The transformation involves scaling the radius by a factor of k=2 and adding π/4 to the angle θ. Hmm, let me think about how to approach this step by step.First, I remember that converting from rectangular coordinates to polar coordinates involves finding the radius r and the angle θ. The radius r can be found using the formula r = sqrt(x² + y²). So, plugging in the values, r = sqrt(5² + 12²) = sqrt(25 + 144) = sqrt(169) = 13. Okay, so r is 13.Next, the angle θ is found using the arctangent function, θ = arctan(y/x). In this case, θ = arctan(12/5). I don't need the exact value in radians or degrees because I'll be working with it symbolically, but it's good to know that θ is the angle whose tangent is 12/5.Now, the transformation involves scaling the radius by k=2, so the new radius becomes kr = 2*13 = 26. The angle is increased by π/4, so the new angle is θ + π/4.To find the new rectangular coordinates, I need to convert back from polar coordinates (26, θ + π/4) to rectangular coordinates. The formulas for conversion are x = r*cos(θ) and y = r*sin(θ). So, the new x-coordinate will be 26*cos(θ + π/4), and the new y-coordinate will be 26*sin(θ + π/4).I remember that there are trigonometric identities for cos(A + B) and sin(A + B). Specifically, cos(A + B) = cos A cos B - sin A sin B, and sin(A + B) = sin A cos B + cos A sin B. Let me apply these identities to both x and y.Starting with the x-coordinate:x = 26*cos(θ + π/4) = 26*(cos θ cos π/4 - sin θ sin π/4)Similarly, for the y-coordinate:y = 26*sin(θ + π/4) = 26*(sin θ cos π/4 + cos θ sin π/4)I know that cos π/4 and sin π/4 are both equal to sqrt(2)/2. So, substituting these values in:x = 26*(cos θ*(sqrt(2)/2) - sin θ*(sqrt(2)/2)) = 26*(sqrt(2)/2)*(cos θ - sin θ) = 13*sqrt(2)*(cos θ - sin θ)Similarly,y = 26*(sin θ*(sqrt(2)/2) + cos θ*(sqrt(2)/2)) = 26*(sqrt(2)/2)*(sin θ + cos θ) = 13*sqrt(2)*(sin θ + cos θ)Now, I need to find the values of cos θ and sin θ from the original rectangular coordinates. Since the original point is (5, 12), and r = 13, we have:cos θ = x/r = 5/13sin θ = y/r = 12/13Substituting these into the expressions for x and y:x = 13*sqrt(2)*(5/13 - 12/13) = 13*sqrt(2)*(-7/13) = -7*sqrt(2)y = 13*sqrt(2)*(12/13 + 5/13) = 13*sqrt(2)*(17/13) = 17*sqrt(2)So, the new rectangular coordinates after the transformation are (-7*sqrt(2), 17*sqrt(2)).Wait, let me double-check my calculations to make sure I didn't make any mistakes. Starting from the beginning: converting (5,12) to polar gives r=13, correct. Then scaling r by 2 gives 26, correct. Adding π/4 to θ, correct. Then using the angle addition formulas, correct. Substituting cos π/4 and sin π/4 as sqrt(2)/2, correct. Then factoring out sqrt(2)/2 and simplifying, correct. Then substituting cos θ = 5/13 and sin θ = 12/13, correct. Calculating x: 13*sqrt(2)*(5/13 - 12/13) = 13*sqrt(2)*(-7/13) = -7*sqrt(2). Similarly for y: 13*sqrt(2)*(12/13 + 5/13) = 13*sqrt(2)*(17/13) = 17*sqrt(2). Looks good.I think that's the correct answer. The rectangular coordinates after the transformation are (-7√2, 17√2).

💡Alright, so I have this trigonometry problem here, and I need to figure out the value of tan frac{y}{2} given some equation. Let me try to break it down step by step.The problem says:cos(x+y) cdot sin x - sin(x+y) cdot cos x = frac{12}{13}And y is an angle in the fourth quadrant. I need to find tan frac{y}{2}.First, I remember there are some trigonometric identities that might help simplify the left side of the equation. Let me recall the sine and cosine addition formulas. Hmm, actually, the expression looks similar to the sine of a difference. Let me think.I remember that sin(A - B) = sin A cos B - cos A sin B. Wait, but in our case, the expression is cos(x+y)sin x - sin(x+y)cos x. Let me write that down:cos(x+y)sin x - sin(x+y)cos xIf I rearrange the terms, it becomes:sin x cos(x+y) - cos x sin(x+y)Which is exactly the form of sin(A - B) where A = x and B = x + y. So, that simplifies to:sin(x - (x + y)) = sin(-y) = -sin ySo, the left side of the equation simplifies to -sin y, and that's equal to frac{12}{13}. Therefore:-sin y = frac{12}{13}Which means:sin y = -frac{12}{13}Okay, so we have sin y = -frac{12}{13}. Since y is in the fourth quadrant, where sine is negative and cosine is positive, that makes sense.Now, I need to find tan frac{y}{2}. To do that, I think I need to find tan y first because there's a half-angle identity that relates tan frac{y}{2} to tan y. Let me recall that identity.The half-angle identity for tangent is:tan frac{y}{2} = frac{sin y}{1 + cos y}Alternatively, it can also be written as:tan frac{y}{2} = frac{1 - cos y}{sin y}Either form might be useful, but let me see which one I can apply here.First, I need to find cos y because I already have sin y = -frac{12}{13}. Since y is in the fourth quadrant, cosine is positive. So, I can use the Pythagorean identity:sin^2 y + cos^2 y = 1Plugging in the value of sin y:left(-frac{12}{13}right)^2 + cos^2 y = 1frac{144}{169} + cos^2 y = 1Subtract frac{144}{169} from both sides:cos^2 y = 1 - frac{144}{169} = frac{25}{169}Taking the square root of both sides:cos y = pm frac{5}{13}But since y is in the fourth quadrant, cosine is positive, so:cos y = frac{5}{13}Great, now I have both sin y and cos y:sin y = -frac{12}{13}cos y = frac{5}{13}Now, let's compute tan y:tan y = frac{sin y}{cos y} = frac{-frac{12}{13}}{frac{5}{13}} = -frac{12}{5}So, tan y = -frac{12}{5}. Now, I need to relate this to tan frac{y}{2}. Let me recall the double-angle identity for tangent, which might help here.The double-angle identity for tangent is:tan y = frac{2 tan frac{y}{2}}{1 - tan^2 frac{y}{2}}So, if I let t = tan frac{y}{2}, then:tan y = frac{2t}{1 - t^2}We know that tan y = -frac{12}{5}, so:frac{2t}{1 - t^2} = -frac{12}{5}Now, let's solve for t. Cross-multiplying:2t cdot 5 = -12 cdot (1 - t^2)10t = -12 + 12t^2Bring all terms to one side:12t^2 - 10t - 12 = 0Hmm, that's a quadratic equation in terms of t. Let me write it as:12t^2 - 10t - 12 = 0To simplify, maybe divide all terms by 2:6t^2 - 5t - 6 = 0Now, let's solve this quadratic equation using the quadratic formula:t = frac{-b pm sqrt{b^2 - 4ac}}{2a}Where a = 6, b = -5, and c = -6.Plugging in the values:t = frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 6 cdot (-6)}}{2 cdot 6}t = frac{5 pm sqrt{25 + 144}}{12}t = frac{5 pm sqrt{169}}{12}sqrt{169} = 13, so:t = frac{5 pm 13}{12}This gives two solutions:1. t = frac{5 + 13}{12} = frac{18}{12} = frac{3}{2}2. t = frac{5 - 13}{12} = frac{-8}{12} = -frac{2}{3}So, tan frac{y}{2} = frac{3}{2} or tan frac{y}{2} = -frac{2}{3}Now, I need to determine which one is the correct value. Since y is in the fourth quadrant, let's think about the value of frac{y}{2}.If y is in the fourth quadrant, it means that y is between frac{3pi}{2} and 2pi (or 270 degrees and 360 degrees). So, if we divide y by 2, then frac{y}{2} will be between frac{3pi}{4} and pi (or 135 degrees and 180 degrees). That places frac{y}{2} in the second quadrant.In the second quadrant, tangent is negative because sine is positive and cosine is negative, so tangent (which is sine over cosine) is negative. Therefore, tan frac{y}{2} must be negative.Looking back at our solutions, we have frac{3}{2} and -frac{2}{3}. Since tan frac{y}{2} must be negative, the correct value is -frac{2}{3}.Let me double-check my steps to make sure I didn't make any mistakes.1. I recognized the trigonometric identity correctly and simplified the equation to -sin y = frac{12}{13}, leading to sin y = -frac{12}{13}.2. Using the Pythagorean identity, I found cos y = frac{5}{13}, which is correct because y is in the fourth quadrant.3. Calculated tan y = -frac{12}{5}, which is correct.4. Applied the double-angle identity for tangent, set up the quadratic equation, and solved for t = tan frac{y}{2}, resulting in two solutions.5. Determined the correct sign based on the quadrant of frac{y}{2}, which is the second quadrant, so tangent is negative.Everything seems to check out. So, I'm confident that the correct answer is -frac{2}{3}.**Final Answer**boxed{C}

💡First, I need to convert the train's speed from kilometers per hour (km/h) to meters per second (m/s) to match the units of the given time.Next, I'll use the formula for distance, which is speed multiplied by time, to find the total distance the train travels while crossing the bridge.Finally, I'll subtract the length of the train from the total distance to determine the length of the bridge.

💡Okay, so I've got this problem here about the cost of books and magazines. Let me try to figure it out step by step. First, the problem says that two books and two magazines cost 26. And then it also says that one book and three magazines cost 27. I need to find out how much one magazine costs. Hmm, okay, so I think I can set up some equations to solve this.Let me assign variables to make it easier. Let's say the cost of one book is B dollars, and the cost of one magazine is M dollars. That way, I can write down the information given in the problem as equations.So, for the first part: two books and two magazines cost 26. That would translate to:2B + 2M = 26And the second part: one book and three magazines cost 27. So that would be:B + 3M = 27Now I have two equations:1) 2B + 2M = 262) B + 3M = 27I need to solve these equations to find the value of M, which is the cost of one magazine. I think I can use the substitution method here. Maybe I can solve one of the equations for one variable and then substitute it into the other equation. Let me try solving the first equation for B.Starting with equation 1:2B + 2M = 26I can subtract 2M from both sides to get:2B = 26 - 2MThen, divide both sides by 2 to solve for B:B = (26 - 2M)/2B = 13 - MOkay, so now I have B expressed in terms of M: B = 13 - MNow, I can substitute this expression for B into equation 2 to find M. Let's do that.Equation 2 is:B + 3M = 27Substituting B = 13 - M into this equation:(13 - M) + 3M = 27Now, let's simplify this:13 - M + 3M = 27Combine like terms:13 + 2M = 27Now, subtract 13 from both sides:2M = 27 - 132M = 14Then, divide both sides by 2:M = 14 / 2M = 7So, the cost of one magazine is 7.Wait, let me double-check to make sure I didn't make any mistakes. If M is 7, then let's find B.From earlier, B = 13 - MSo, B = 13 - 7 = 6So, one book costs 6 and one magazine costs 7.Let's verify these values with the original equations.First equation: 2B + 2M = 262*6 + 2*7 = 12 + 14 = 26. That checks out.Second equation: B + 3M = 276 + 3*7 = 6 + 21 = 27. That also checks out.Okay, so I think I did it right. The cost of one magazine is 7.

💡Alright, so I've got this expression here: 7x plus 3 minus 2x plus 15 plus y. Hmm, okay, let me see. I think the first thing I need to do is simplify this expression by combining like terms. Like terms are terms that have the same variable raised to the same power, right? So, in this case, I have terms with x, a constant, and a y term.Let me start by looking at the x terms. I have 7x and then minus 2x. So, 7x minus 2x should be straightforward. If I have 7 of something and I take away 2, I'm left with 5. So, 7x minus 2x equals 5x. Okay, that seems simple enough.Next, I need to look at the constant terms. I have a plus 3 and a plus 15. Adding those together, 3 plus 15 equals 18. So, the constants simplify to 18.Now, what about the y term? I see a plus y at the end of the expression. Since there are no other y terms to combine with, I guess it just stays as it is.Putting it all together, I have 5x from the x terms, 18 from the constants, and y remains as is. So, the simplified expression should be 5x plus y plus 18.Wait a second, does the order matter here? I think in algebra, the order of terms doesn't affect the value of the expression, so 5x plus y plus 18 is the same as 5x plus 18 plus y or even y plus 5x plus 18. But I think it's conventional to write the terms in a certain order, usually starting with the variables in alphabetical order and then the constants. So, maybe writing it as 5x plus y plus 18 is the most standard way.Let me double-check my work to make sure I didn't miss anything. I combined 7x and minus 2x to get 5x, which seems right. Then, I added 3 and 15 to get 18, which is correct. And the y term didn't have any like terms to combine with, so it stays as y. Yeah, I think that's all there is to it.I guess another way to look at it is by grouping the like terms together before simplifying. So, grouping 7x and minus 2x gives me 5x, and grouping 3 and 15 gives me 18. Then, adding y to that gives me the final simplified expression. Yep, that makes sense.Is there any possibility I could have made a mistake in my calculations? Let me verify the arithmetic. 7x minus 2x is indeed 5x because 7 minus 2 is 5. And 3 plus 15 is definitely 18. So, no, I don't think I made any errors there.What if I approached it differently? Maybe by rearranging the terms to group the x terms, y terms, and constants separately. So, starting with 7x minus 2x, which is 5x, then y, and then 3 plus 15, which is 18. So, again, I end up with 5x plus y plus 18. That seems consistent.I think I'm confident that the simplified form of the expression is 5x plus y plus 18. There's no further simplification needed because there are no like terms left to combine. The x and y terms are different variables, so they can't be combined, and 18 is just a constant. So, yeah, that's as simplified as it gets.Just to recap, the original expression was 7x plus 3 minus 2x plus 15 plus y. By combining the x terms and the constants separately, and leaving the y term as is, I arrived at the simplified expression 5x plus y plus 18. I don't see any other steps or considerations needed here. It seems pretty straightforward.I wonder if there's a way to represent this visually or with some kind of diagram to make it clearer. Maybe by using color coding or underlining the like terms as I combine them. But I think for the purposes of this problem, just following the algebraic steps is sufficient.Another thing I could do is plug in some numbers to test if the original expression and the simplified expression give the same result. For example, let's say x is 1 and y is 2. Plugging into the original expression: 7(1) + 3 - 2(1) + 15 + 2 equals 7 + 3 - 2 + 15 + 2, which is 7 + 3 is 10, minus 2 is 8, plus 15 is 23, plus 2 is 25. Now, plugging into the simplified expression: 5(1) + 2 + 18 equals 5 + 2 + 18, which is 5 + 2 is 7, plus 18 is 25. So, both expressions give the same result, which is a good sign that my simplification is correct.Let me try another set of numbers to be thorough. Let's say x is 3 and y is 4. Original expression: 7(3) + 3 - 2(3) + 15 + 4 equals 21 + 3 - 6 + 15 + 4. Calculating step by step: 21 + 3 is 24, minus 6 is 18, plus 15 is 33, plus 4 is 37. Simplified expression: 5(3) + 4 + 18 equals 15 + 4 + 18, which is 15 + 4 is 19, plus 18 is 37. Again, both expressions give the same result. This further confirms that my simplification is accurate.I think I've covered all the bases here. I've combined like terms correctly, verified my work with arithmetic checks, and even tested the expressions with specific values to ensure they're equivalent. So, I'm pretty sure that the simplified form of the given expression is indeed 5x plus y plus 18.

💡Alright, so I have this equation to solve: ((x^2 + x)^2 + (x^2 + x) - 6 = 0). Hmm, it looks a bit complicated because of the ((x^2 + x)) terms. Maybe I can simplify it somehow. Let me think... Oh, substitution! Yeah, if I let (y = x^2 + x), then the equation becomes (y^2 + y - 6 = 0). That seems easier to handle. Okay, so now I have a quadratic equation in terms of (y). I can try to factor this or use the quadratic formula. Let me see if it factors nicely.Looking at (y^2 + y - 6), I need two numbers that multiply to (-6) and add up to (1). Hmm, 3 and (-2) work because (3 times (-2) = -6) and (3 + (-2) = 1). So, the equation factors to ((y + 3)(y - 2) = 0). Setting each factor equal to zero gives me the possible values for (y):1. (y + 3 = 0) leads to (y = -3)2. (y - 2 = 0) leads to (y = 2)Now, I need to substitute back (y = x^2 + x) to find the values of (x).Starting with (y = -3):[x^2 + x = -3]Let me rearrange this to form a standard quadratic equation:[x^2 + x + 3 = 0]To solve this, I'll use the quadratic formula:[x = frac{-b pm sqrt{b^2 - 4ac}}{2a}]Here, (a = 1), (b = 1), and (c = 3). Plugging these in:[x = frac{-1 pm sqrt{1^2 - 4(1)(3)}}{2(1)} = frac{-1 pm sqrt{1 - 12}}{2} = frac{-1 pm sqrt{-11}}{2}]Oh, the discriminant is negative ((-11)), which means there are no real solutions for this case. So, (y = -3) doesn't give us any real (x) values.Moving on to (y = 2):[x^2 + x = 2]Again, rearranging to form a standard quadratic equation:[x^2 + x - 2 = 0]Let me try factoring this. I need two numbers that multiply to (-2) and add up to (1). That would be (2) and (-1):[(x + 2)(x - 1) = 0]Setting each factor equal to zero gives:1. (x + 2 = 0) leads to (x = -2)2. (x - 1 = 0) leads to (x = 1)So, the solutions for (x) are (-2) and (1). Wait, let me double-check my factoring for (x^2 + x - 2). If I expand ((x + 2)(x - 1)), I get (x^2 - x + 2x - 2 = x^2 + x - 2), which matches. Okay, that seems correct.And for the case where (y = -3), I correctly identified that the discriminant is negative, so no real solutions there. So, putting it all together, the real solutions to the original equation are (x = -2) and (x = 1).