Given that tan x=(1)/(sqrt3) and cos y=(sqrt5)/(5), and that angles x and y are both in Quadrant I, find the exact value of cos(x+y), in simplest radical form. Answer:

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Given that  tan x=(1)/(sqrt3) and  cos y=(sqrt5)/(5), and that angles  x and  y are both in Quadrant I, find the exact value of  cos(x+y), in simplest radical form. Answer:

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Use Cosine Addition Formula: To find cos(x+y), we can use the cosine addition formula: cos(x+y) = cos(x)cos(y) - sin(x)sin(y). We already have cos(y), but we need to find cos(x) and sin(x) as well as sin(y).Find Cos(x) and Sin(x): Since tan x = 1/sqrt(3), we can create a right triangle where the opposite side is 1 and the adjacent side is sqrt(3). The hypotenuse, using the Pythagorean theorem, is sqrt(1^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2. Therefore, cos(x) = adjacent/hypotenuse = sqrt(3)/2.Find Sin(y): Similarly, sin(x) can be found using the right triangle, where sin(x) = opposite/hypotenuse = 1/2.Substitute Values: To find sin(y), we use the Pythagorean identity sin^2(y) + cos^2(y) = 1. We know cos(y) = sqrt(5)/5, so cos^2(y) = (sqrt(5)/5)^2 = 5/25 = 1/5. Therefore, sin^2(y) = 1 - 1/5 = 4/5. Since y is in Quadrant I, sin(y) is positive, so sin(y) = sqrt(4/5) = 2/sqrt(5) = 2sqrt(5)/5 after rationalizing the denominator.Simplify Expression: Now we have all the necessary values: cos(x) = sqrt(3)/2, sin(x) = 1/2, cos(y) = sqrt(5)/5, and sin(y) = 2sqrt(5)/5. We can substitute these into the cosine addition formula: cos(x+y) = (sqrt(3)/2)(sqrt(5)/5) - (1/2)(2sqrt(5)/5).Final Answer: Simplify the expression: cos(x+y) = (sqrt(3)sqrt(5)/10) - (sqrt(5)/5) = (sqrt(15)/10) - (2sqrt(5)/10) = (sqrt(15) - 2sqrt(5))/10.The expression (sqrt(15) - 2sqrt(5))/10 is already in simplest radical form, so this is our final answer.

Given that $\tan x=\frac{1}{\sqrt{3}}$ and $\cos y=\frac{\sqrt{5}}{5}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\cos (x+y)$ , in simplest radical form. $\newline$ Answer:

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Given that $\tan x=\frac{1}{\sqrt{3}}$ and $\cos y=\frac{\sqrt{5}}{5}$ , and that angles $x$ and $y$ are both in Quadrant I, find the exact value of $\cos (x+y)$ , in simplest radical form. $\newline$ Answer: