Given that cos x=(sqrt5)/(4) and sin y=(1)/(4), 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  cos x=(sqrt5)/(4) and  sin y=(1)/(4), 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 Trigonometric Values: We are given that cos x = (sqrt5)/(4) and sin y = (1)/(4), and we need to find cos(x-y). We can use the cosine difference identity, which states that cos(x-y) = cos x * cos y + sin x * sin y. First, we need to find cos y and sin x.Find cos y: Since sin y = (1)/(4) and y is in Quadrant I, where all trigonometric functions are positive, we can find cos y using the Pythagorean identity sin²y + cos²y = 1. We calculate cos y as follows: cos²y = 1 - sin²y cos²y = 1 - (1/4)² cos²y = 1 - 1/16 cos²y = 15/16 cos y = sqrt(15/16) cos y = sqrt(15)/4Find sin x: Similarly, we can find sin x using the Pythagorean identity sin²x + cos²x = 1. We calculate sin x as follows: sin²x = 1 - cos²x sin²x = 1 - (sqrt5/4)² sin²x = 1 - 5/16 sin²x = 11/16 sin x = sqrt(11/16) sin x = sqrt(11)/4Calculate cos(x-y): Now that we have cos y = sqrt(15)/4 and sin x = sqrt(11)/4, we can substitute these values into the cosine difference identity: cos(x-y) = cos x * cos y + sin x * sin y cos(x-y) = (sqrt5/4) * (sqrt(15)/4) + (sqrt(11)/4) * (1/4) cos(x-y) = (sqrt(5) * sqrt(15))/(4 * 4) + (sqrt(11))/(4 * 4) cos(x-y) = (sqrt(75) + sqrt(11))/16Simplify sqrt(75): We can simplify sqrt(75) by factoring out the perfect square: sqrt(75) = sqrt(25 * 3) sqrt(75) = sqrt(25) * sqrt(3) sqrt(75) = 5 * sqrt(3)Substitute and Simplify: Substitute the simplified form of sqrt(75) back into the expression for cos(x-y): cos(x-y) = (5 * sqrt(3) + sqrt(11))/16 This is the exact value of cos(x-y) in simplest radical form.

Given that $\cos x=\frac{\sqrt{5}}{4}$ and $\sin y=\frac{1}{4}$ , 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 $\cos x=\frac{\sqrt{5}}{4}$ and $\sin y=\frac{1}{4}$ , 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: