Given that tan A=(1)/(2) and sin B=(6)/(sqrt85), and that angles A and B are both in Quadrant I, find the exact value of cos(A+B), in simplest radical form. Answer:

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Given that  tan A=(1)/(2) and  sin B=(6)/(sqrt85), and that angles  A and  B are both in Quadrant I, find the exact value of  cos(A+B), in simplest radical form. Answer:

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Find cos A: Use the Pythagorean identity to find cos A. Since tan A = 1/2, we can represent this as opposite/adjacent in a right triangle, where the opposite side is 1 and the adjacent side is 2. To find the hypotenuse (h), we use the Pythagorean theorem: h^2 = 1^2 + 2^2. Calculate the hypotenuse: h^2 = 1 + 4 h^2 = 5 h = sqrt(5) Now, find cos A using the adjacent side and hypotenuse: cos A = adjacent/hypotenuse = 2/sqrt(5). Rationalize the denominator: cos A = 2/sqrt(5) * sqrt(5)/sqrt(5) = 2sqrt(5)/5.Find cos B: Use the Pythagorean identity to find cos B. Since sin B = 6/sqrt(85), we can represent this as opposite/hypotenuse in a right triangle, where the opposite side is 6 and the hypotenuse is sqrt(85). To find the adjacent side (a), we use the Pythagorean theorem: a^2 = hypotenuse^2 - opposite^2. Calculate the adjacent side: a^2 = (sqrt(85))^2 - 6^2 a^2 = 85 - 36 a^2 = 49 a = sqrt(49) a = 7 Now, find cos B using the adjacent side and hypotenuse: cos B = adjacent/hypotenuse = 7/sqrt(85). Rationalize the denominator: cos B = 7/sqrt(85) * sqrt(85)/sqrt(85) = 7sqrt(85)/85.Find cos(A+B): Use the angle sum formula for cosine to find cos(A+B). The formula for cos(A+B) is cos A * cos B - sin A * sin B. We already have cos A = 2sqrt(5)/5 and cos B = 7sqrt(85)/85. To find sin A, we use the definition of tan A = sin A / cos A. Since tan A = 1/2 and cos A = 2sqrt(5)/5, we can solve for sin A: sin A = tan A * cos A = (1/2) * (2sqrt(5)/5) = sqrt(5)/5. Now, substitute the values into the formula: cos(A+B) = (2sqrt(5)/5) * (7sqrt(85)/85) - (sqrt(5)/5) * (6/sqrt(85)).Simplify expression: Simplify the expression for cos(A+B). First, multiply the cosines: (2sqrt(5)/5) * (7sqrt(85)/85) = (14sqrt(425))/425. Then, multiply the sines: (sqrt(5)/5) * (6/sqrt(85)) = (6sqrt(5))/(5sqrt(85)). Rationalize the denominator of the second term: (6sqrt(5))/(5sqrt(85)) * (sqrt(85)/sqrt(85)) = (6sqrt(425))/(425). Now, subtract the second term from the first term: (14sqrt(425))/425 - (6sqrt(425))/(425) = (8sqrt(425))/425.Check for further simplification: Check if the expression can be simplified further. The term sqrt(425) can be simplified since 425 is not a prime number. Factor 425 to find perfect squares: 425 = 25 * 17. Now, simplify sqrt(425): sqrt(425) = sqrt(25 * 17) = 5sqrt(17). Substitute this back into the expression for cos(A+B): cos(A+B) = (8 * 5sqrt(17))/425. Simplify the fraction: cos(A+B) = 40sqrt(17)/425. Since 40 and 425 have a common factor of 5, divide both numerator and denominator by 5: cos(A+B) = 8sqrt(17)/85.

Given that $\tan A=\frac{1}{2}$ and $\sin B=\frac{6}{\sqrt{85}}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\cos (A+B)$ , in simplest radical form. $\newline$ Answer:

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Given that $\tan A=\frac{1}{2}$ and $\sin B=\frac{6}{\sqrt{85}}$ , and that angles $A$ and $B$ are both in Quadrant I, find the exact value of $\cos (A+B)$ , in simplest radical form. $\newline$ Answer: