Given that tan x=(3)/(2) and cos y=(6)/(sqrt37), and that angles x and y are both in Quadrant I, find the exact value of sin(x-y), in simplest radical form. Answer:

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Given that  tan x=(3)/(2) and  cos y=(6)/(sqrt37), and that angles  x and  y are both in Quadrant I, find the exact value of  sin(x-y), in simplest radical form. Answer:

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Given values: We are given tan x = 3/2 and cos y = 6/sqrt(37). Since tan x = sin x / cos x, we can create a right triangle with opposite side 3, adjacent side 2, and hypotenuse sqrt(3^2 + 2^2) = sqrt(13) to find sin x and cos x.Finding sin x: Using the Pythagorean theorem, sin x = opposite/hypotenuse = 3/sqrt(13). We rationalize the denominator to get sin x = 3sqrt(13)/13.Finding sin y: We are also given cos y = 6/sqrt(37). To find sin y, we use the Pythagorean identity sin^2 y + cos^2 y = 1. Substituting cos y, we get sin^2 y + (6/sqrt(37))^2 = 1.Using sine difference formula: Solving for sin y, we have sin^2 y = 1 - (36/37). This simplifies to sin^2 y = (37/37) - (36/37) = 1/37. Taking the square root, sin y = sqrt(1/37) = 1/sqrt(37), and we rationalize the denominator to get sin y = sqrt(37)/37.Simplify the expression: Now we use the sine difference formula: sin(x-y) = sin x * cos y - cos x * sin y. We substitute the values we found: sin(x-y) = (3sqrt(13)/13) * (6/sqrt(37)) - (2/sqrt(13)) * (sqrt(37)/37).Further simplification: Simplify the expression: sin(x-y) = (18sqrt(13)/sqrt(37)/13) - (2sqrt(37)/sqrt(13)/37). This simplifies to sin(x-y) = (18sqrt(13)*sqrt(37))/(13*sqrt(37)) - (2sqrt(37)*sqrt(13))/(37*sqrt(13)).Combine the terms: Further simplification gives us sin(x-y) = (18sqrt(481))/(13*37) - (2sqrt(481))/(13*37). Since the denominators are the same, we can combine the terms.Final simplification: Combining the terms, we get sin(x-y) = (18sqrt(481) - 2sqrt(481))/(13*37). This simplifies to sin(x-y) = 16sqrt(481)/(13*37).Finally, we simplify the expression by dividing both numerator and denominator by their greatest common divisor, which is 1 in this case. So, the expression is already in its simplest form: sin(x-y) = 16sqrt(481)/(13*37).

Given that $\tan x=\frac{3}{2}$ and $\cos y=\frac{6}{\sqrt{37}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x-y)$ , in simplest radical form. $\newline$ Answer:

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Given that $\tan x=\frac{3}{2}$ and $\cos y=\frac{6}{\sqrt{37}}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\sin (x-y)$ , in simplest radical form. $\newline$ Answer: