Given that cos x=(3)/(sqrt10) and tan y=(3)/(2), 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  cos x=(3)/(sqrt10) and  tan y=(3)/(2), 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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Find sin x: Use the Pythagorean identity to find sin x. Since cos x = 3/sqrt(10), we can find sin x using the identity sin^2(x) + cos^2(x) = 1. sin^2(x) = 1 - cos^2(x) sin^2(x) = 1 - (3/sqrt(10))^2 sin^2(x) = 1 - 9/10 sin^2(x) = 10/10 - 9/10 sin^2(x) = 1/10 sin(x) = sqrt(1/10) sin(x) = sqrt(1)/sqrt(10) sin(x) = 1/sqrt(10)Find sin y: Use the definition of tangent to find sin y. Since tan y = 3/2, and tan y = sin y / cos y, we can find sin y using the Pythagorean identity sin^2(y) + cos^2(y) = 1. Let's find cos y first: tan^2(y) + 1 = sec^2(y) (3/2)^2 + 1 = 1/cos^2(y) 9/4 + 1 = 1/cos^2(y) 13/4 = 1/cos^2(y) cos^2(y) = 4/13 cos(y) = sqrt(4/13) cos(y) = 2/sqrt(13) Now, we can find sin y: sin y = tan y * cos y sin y = (3/2) * (2/sqrt(13)) sin y = 3/sqrt(13)Find sin(x-y): Use the angle difference identity for sine to find sin(x-y). The identity is sin(x-y) = sin x * cos y - cos x * sin y. sin(x-y) = (1/sqrt(10)) * (2/sqrt(13)) - (3/sqrt(10)) * (3/sqrt(13)) sin(x-y) = (2/sqrt(130)) - (9/sqrt(130)) sin(x-y) = (2 - 9)/sqrt(130) sin(x-y) = -7/sqrt(130)Rationalize sin(x-y): Rationalize the denominator of sin(x-y). To rationalize the denominator, multiply the numerator and the denominator by sqrt(130). sin(x-y) = (-7/sqrt(130)) * (sqrt(130)/sqrt(130)) sin(x-y) = -7*sqrt(130)/130

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