Given that cos x=(sqrt5)/(5) and sin y=(1)/(3), 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=(sqrt5)/(5) and  sin y=(1)/(3), 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 Sine Addition Formula: To find sin(x+y), we will use the sine addition formula: sin(x+y) = sin(x)cos(y) + cos(x)sin(y). First, we need to find sin(x) and cos(y).Find sin(x): Since x is in Quadrant I, sin(x) is positive. We can find sin(x) using the Pythagorean identity: sin^2(x) + cos^2(x) = 1. We know cos(x) = (sqrt5)/(5), so we can solve for sin(x). sin^2(x) = 1 - cos^2(x) sin^2(x) = 1 - [(sqrt5)/(5)]^2 sin^2(x) = 1 - [5/25] sin^2(x) = 1 - [1/5] sin^2(x) = [5/5] - [1/5] sin^2(x) = [4/5] sin(x) = sqrt([4/5]) sin(x) = (2sqrt(5))/(5)Find cos(y): Now we need to find cos(y). Since y is in Quadrant I, cos(y) is also positive. Using the Pythagorean identity again: sin^2(y) + cos^2(y) = 1. We know sin(y) = (1)/(3), so we can solve for cos(y). cos^2(y) = 1 - sin^2(y) cos^2(y) = 1 - [(1)/(3)]^2 cos^2(y) = 1 - [1/9] cos^2(y) = [9/9] - [1/9] cos^2(y) = [8/9] cos(y) = sqrt([8/9]) cos(y) = (2sqrt(2))/(3)Calculate sin(x+y): Now that we have sin(x) and cos(y), we can use the sine addition formula to find sin(x+y). sin(x+y) = sin(x)cos(y) + cos(x)sin(y) sin(x+y) = [(2sqrt(5))/(5)] * [(2sqrt(2))/(3)] + [(sqrt5)/(5)] * [(1)/(3)] sin(x+y) = [(4sqrt(10))/(15)] + [(sqrt5)/(15)] sin(x+y) = [(4sqrt(10) + sqrt5)/(15)]

Given that $\cos x=\frac{\sqrt{5}}{5}$ and $\sin y=\frac{1}{3}$ , 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 $\cos x=\frac{\sqrt{5}}{5}$ and $\sin y=\frac{1}{3}$ , 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: