Given that sin x=(3)/(sqrt58) and tan y=1, 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  sin x=(3)/(sqrt58) and  tan y=1, 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 Information: We are given sin x = 3/√58 and tan y = 1. Since both angles are in Quadrant I, all trigonometric functions are positive. We need to find cos(x-y). To do this, we will use the cosine difference identity: cos(x-y) = cos x cos y + sin x sin y. First, we need to find cos x and cos y.Finding cos x: To find cos x, we use the Pythagorean identity sin^2 x + cos^2 x = 1. We have sin x = 3/√58, so sin^2 x = (3/√58)^2 = 9/58. Therefore, cos^2 x = 1 - sin^2 x = 1 - 9/58 = 49/58. Since x is in Quadrant I, cos x is positive, so cos x = √(49/58) = 7/√58.Finding cos y: For angle y, we have tan y = 1. In Quadrant I, this means that sin y = cos y because tan y = sin y / cos y. Since tan y = 1, sin y = cos y. To find the value of sin y and cos y, we use the Pythagorean identity sin^2 y + cos^2 y = 1. Since sin y = cos y, we have 2sin^2 y = 1. Therefore, sin^2 y = 1/2, and sin y = cos y = √(1/2) = √2/2.Using Cosine Difference Identity: Now that we have cos x = 7/√58 and cos y = √2/2, and we know sin x = 3/√58 and sin y = √2/2, we can use the cosine difference identity to find cos(x-y). So, cos(x-y) = cos x cos y + sin x sin y = (7/√58)(√2/2) + (3/√58)(√2/2).Simplifying Expression: We simplify the expression: cos(x-y) = (7/√58)(√2/2) + (3/√58)(√2/2) = (7√2/2√58) + (3√2/2√58) = (7√2 + 3√2)/(2√58) = 10√2/(2√58) = 5√2/√58.Rationalizing Denominator: To rationalize the denominator, we multiply the numerator and the denominator by √58: cos(x-y) = (5√2/√58) * (√58/√58) = (5√2√58)/(√58√58) = (5√116)/58.Simplifying Square Root: We simplify the square root in the numerator: √116 = √(4*29) = 2√29. So, cos(x-y) = (5*2√29)/58 = 10√29/58.Final Simplification: Finally, we simplify the fraction by dividing both the numerator and the denominator by 2: cos(x-y) = (10√29/58) / (2/2) = 5√29/29.

Given that $\sin x=\frac{3}{\sqrt{58}}$ and $\tan y=1$ , 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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