Given that sin x=(1)/(sqrt2) and cos y=(3)/(sqrt10), 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  sin x=(1)/(sqrt2) and  cos y=(3)/(sqrt10), 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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Apply Sine Subtraction Formula: Use the sine subtraction formula: sin(x - y) = sin(x)cos(y) - cos(x)sin(y). We already know sin(x) = 1/sqrt(2) and cos(y) = 3/sqrt(10). Next, we need to find cos(x) and sin(y).Find cos(x) and sin(y): Since x is in Quadrant I and sin(x) = 1/sqrt(2), we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to find cos(x). cos^2(x) = 1 - sin^2(x) cos^2(x) = 1 - (1/sqrt(2))^2 cos^2(x) = 1 - 1/2 cos^2(x) = 1/2 cos(x) = sqrt(1/2) = 1/sqrt(2), since cos(x) is positive in Quadrant I.Substitute Values: Similarly, since y is in Quadrant I and cos(y) = 3/sqrt(10), we use the Pythagorean identity sin^2(y) + cos^2(y) = 1 to find sin(y). sin^2(y) = 1 - cos^2(y) sin^2(y) = 1 - (3/sqrt(10))^2 sin^2(y) = 1 - 9/10 sin^2(y) = 1/10 sin(y) = sqrt(1/10) = 1/sqrt(10), since sin(y) is positive in Quadrant I.Simplify Expression: Now we can substitute the values into the sine subtraction formula: sin(x - y) = sin(x)cos(y) - cos(x)sin(y) sin(x - y) = (1/sqrt(2))(3/sqrt(10)) - (1/sqrt(2))(1/sqrt(10))Rationalize Denominator: Simplify the expression: sin(x - y) = 3/(sqrt(2)sqrt(10)) - 1/(sqrt(2)sqrt(10)) sin(x - y) = (3 - 1)/(sqrt(2)sqrt(10)) sin(x - y) = 2/(sqrt(2)sqrt(10))Rationalize the denominator: sin(x - y) = 2/(sqrt(2)sqrt(10)) * (sqrt(2)sqrt(10))/(sqrt(2)sqrt(10)) sin(x - y) = 2sqrt(2)sqrt(10)/(2*10) sin(x - y) = sqrt(2)sqrt(10)/10 sin(x - y) = sqrt(20)/10 sin(x - y) = sqrt(4*5)/10 sin(x - y) = (2sqrt(5))/10 sin(x - y) = sqrt(5)/5

Given that $\sin x=\frac{1}{\sqrt{2}}$ and $\cos y=\frac{3}{\sqrt{10}}$ , 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 $\sin x=\frac{1}{\sqrt{2}}$ and $\cos y=\frac{3}{\sqrt{10}}$ , 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: