Given that tan x=(2)/(sqrt5) and sin y=(4)/(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=(2)/(sqrt5) and  sin y=(4)/(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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Find cos x: We know that tan x = 2/sqrt(5). To find cos x, we use the identity tan^2 x + 1 = sec^2 x, which can be rewritten as 1/cos^2 x. We can then solve for cos x. tan^2 x = (2/sqrt(5))^2 = 4/5 1 + tan^2 x = 1 + 4/5 = 9/5 Therefore, sec^2 x = 9/5, which means cos^2 x = 5/9. Since x is in Quadrant I, cos x is positive, so cos x = sqrt(5/9) = sqrt(5)/3.Find cos y: We are given sin y = 4/5. To find cos y, we use the Pythagorean identity sin^2 y + cos^2 y = 1. We can then solve for cos y. sin^2 y = (4/5)^2 = 16/25 1 - sin^2 y = 1 - 16/25 = 9/25 Therefore, cos^2 y = 9/25, which means cos y = sqrt(9/25) = 3/5. Since y is in Quadrant I, cos y is positive, so cos y = 3/5.Calculate cos(x+y): Now we need to find cos(x+y). We use the angle sum identity for cosine, which is cos(x+y) = cos x * cos y - sin x * sin y. We already have cos x = sqrt(5)/3 and cos y = 3/5. To find sin x, we use the Pythagorean identity sin^2 x + cos^2 x = 1. cos^2 x = (sqrt(5)/3)^2 = 5/9 1 - cos^2 x = 1 - 5/9 = 4/9 Therefore, sin^2 x = 4/9, which means sin x = sqrt(4/9) = 2/3. Since x is in Quadrant I, sin x is positive, so sin x = 2/3.Now we can calculate cos(x+y) using the values we have found: cos(x+y) = (sqrt(5)/3) * (3/5) - (2/3) * (4/5) cos(x+y) = (sqrt(5)/5) - (8/15) To combine these fractions, we need a common denominator, which is 15. cos(x+y) = (3*sqrt(5)/15) - (8/15) cos(x+y) = (3*sqrt(5) - 8)/15

Given that $\tan x=\frac{2}{\sqrt{5}}$ and $\sin y=\frac{4}{5}$ , 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 $\tan x=\frac{2}{\sqrt{5}}$ and $\sin y=\frac{4}{5}$ , 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: