Given that tan A=2 and sin B=(5)/(sqrt41), and that angles A and B are both in Quadrant I, find the exact value of sin(A-B), in simplest radical form. Answer:

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Given that  tan A=2 and  sin B=(5)/(sqrt41), and that angles  A and  B are both in Quadrant I, find the exact value of  sin(A-B), in simplest radical form. Answer:

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Apply Angle Subtraction Formula: We will use the angle subtraction formula for sine, which is sin(A-B) = sin(A)cos(B) - cos(A)sin(B). To use this formula, we need to find sin(A), cos(A), and cos(B).Find sin(A), cos(A), cos(B): Since tan A = 2 and we are in Quadrant I, we can create a right triangle where the opposite side to angle A is 2 and the adjacent side is 1. Using the Pythagorean theorem, the hypotenuse is sqrt(1^2 + 2^2) = sqrt(5). Therefore, sin(A) = opposite/hypotenuse = 2/sqrt(5). To rationalize, we multiply by sqrt(5)/sqrt(5) to get sin(A) = 2sqrt(5)/5.Use Pythagorean Identity for sin(B): Using the same triangle, cos(A) = adjacent/hypotenuse = 1/sqrt(5). Rationalizing, we get cos(A) = sqrt(5)/5.Calculate cos(B): We are given sin B = 5/sqrt(41). To find cos(B), we use the Pythagorean identity sin^2(B) + cos^2(B) = 1. Substituting sin B, we get (5/sqrt(41))^2 + cos^2(B) = 1. Simplifying, we get 25/41 + cos^2(B) = 1.Substitute into Angle Subtraction Formula: Subtract 25/41 from both sides to solve for cos^2(B): cos^2(B) = 1 - 25/41. This simplifies to cos^2(B) = (41/41) - (25/41) = 16/41. Taking the square root, cos(B) = sqrt(16/41) = 4/sqrt(41). Since B is in Quadrant I, cos(B) is positive, so cos(B) = 4/sqrt(41). Rationalizing, we get cos(B) = 4sqrt(41)/41.Combine Terms: Now we have sin(A) = 2sqrt(5)/5, cos(A) = sqrt(5)/5, and cos(B) = 4sqrt(41)/41. We can substitute these into the angle subtraction formula: sin(A-B) = (2sqrt(5)/5)(4sqrt(41)/41) - (sqrt(5)/5)(5/sqrt(41)).Rewrite Fraction: Multiplying the terms, we get sin(A-B) = (8sqrt(205)/205) - (5sqrt(5)/41). To combine these terms, we need a common denominator, which is 205.Subtract Fractions: We rewrite (5sqrt(5)/41) as (25sqrt(5)/205) to have the same denominator. Now, sin(A-B) = (8sqrt(205)/205) - (25sqrt(5)/205).Final Result: Subtracting the fractions, we get sin(A-B) = (8sqrt(205) - 25sqrt(5))/205. This is the exact value of sin(A-B) in simplest radical form.

Given that $\tan A=2$ and $\sin B=\frac{5}{\sqrt{41}}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\sin (A-B)$ , in simplest radical form. $\newline$ Answer:

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Given that $\tan A=2$ and $\sin B=\frac{5}{\sqrt{41}}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\sin (A-B)$ , in simplest radical form. $\newline$ Answer: