Given that tan x=sqrt3 and tan y=(3)/(sqrt7), 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=sqrt3 and  tan y=(3)/(sqrt7), 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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Use Angle Subtraction Formula: We will use the angle subtraction formula for sine, which is sin(x-y) = sin(x)cos(y) - cos(x)sin(y). To find sin(x) and cos(x), we use the fact that tan x = sin x / cos x. Since tan x = sqrt(3), we can create a right triangle where the opposite side to angle x is sqrt(3) and the adjacent side is 1, because tan x = opposite/adjacent. Then, using the Pythagorean theorem, we find the hypotenuse.Find Triangle Sides: For angle x, we have tan x = sqrt(3)/1, so the sides of the triangle are opposite = sqrt(3), adjacent = 1. Using the Pythagorean theorem, hypotenuse^2 = opposite^2 + adjacent^2, we get hypotenuse^2 = (sqrt(3))^2 + 1^2 = 3 + 1 = 4. Therefore, the hypotenuse is sqrt(4) = 2.Calculate sin x and cos x: Now we can find sin x and cos x. Since sin x = opposite/hypotenuse, we have sin x = sqrt(3)/2. And since cos x = adjacent/hypotenuse, we have cos x = 1/2.Find Triangle Sides: Next, we find sin y and cos y using the same method. We have tan y = 3/sqrt(7), so we can create a right triangle where the opposite side to angle y is 3 and the adjacent side is sqrt(7). Again, using the Pythagorean theorem, we find the hypotenuse.Calculate sin y and cos y: For angle y, we have tan y = 3/sqrt(7), so the sides of the triangle are opposite = 3, adjacent = sqrt(7). Using the Pythagorean theorem, hypotenuse^2 = opposite^2 + adjacent^2, we get hypotenuse^2 = 3^2 + (sqrt(7))^2 = 9 + 7 = 16. Therefore, the hypotenuse is sqrt(16) = 4.Substitute Values: Now we can find sin y and cos y. Since sin y = opposite/hypotenuse, we have sin y = 3/4. And since cos y = adjacent/hypotenuse, we have cos y = sqrt(7)/4.Simplify Expression: We can now substitute the values of sin x, cos x, sin y, and cos y into the angle subtraction formula for sine: sin(x-y) = sin(x)cos(y) - cos(x)sin(y) = (sqrt(3)/2)(sqrt(7)/4) - (1/2)(3/4).Combine Terms: Simplify the expression: sin(x-y) = (sqrt(3) * sqrt(7))/(2 * 4) - (1 * 3)/(2 * 4) = (sqrt(21)/8) - (3/8).Combine the terms to get the final answer: sin(x-y) = (sqrt(21) - 3)/8.

Given that $\tan x=\sqrt{3}$ and $\tan y=\frac{3}{\sqrt{7}}$ , 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 $\tan x=\sqrt{3}$ and $\tan y=\frac{3}{\sqrt{7}}$ , 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: