If sin A=(40)/(41) and cos B=(20)/(29) and angles A and B are in Quadrant I, find the value of tan(A-B). Answer:

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If  sin A=(40)/(41) and  cos B=(20)/(29) and angles A and B are in Quadrant I, find the value of  tan(A-B). Answer:

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Find cos(A): Use the Pythagorean identity to find cos(A). Since sin(A) = 40/41 and A is in Quadrant I, we can use the Pythagorean identity sin^2(A) + cos^2(A) = 1 to find cos(A). cos^2(A) = 1 - sin^2(A) cos^2(A) = 1 - (40/41)^2 cos^2(A) = 1 - 1600/1681 cos^2(A) = (1681/1681) - (1600/1681) cos^2(A) = 81/1681 cos(A) = sqrt(81/1681) cos(A) = 9/41Find sin(B): Use the Pythagorean identity to find sin(B). Since cos(B) = 20/29 and B is in Quadrant I, we can use the Pythagorean identity sin^2(B) + cos^2(B) = 1 to find sin(B). sin^2(B) = 1 - cos^2(B) sin^2(B) = 1 - (20/29)^2 sin^2(B) = 1 - 400/841 sin^2(B) = (841/841) - (400/841) sin^2(B) = 441/841 sin(B) = sqrt(441/841) sin(B) = 21/29Find tan(A-B): Use the angle subtraction formula for tangent to find tan(A-B). The formula for tan(A-B) is (tan(A) - tan(B)) / (1 + tan(A)tan(B)). We already have sin(A) and cos(A), and sin(B) and cos(B), so we can find tan(A) and tan(B). tan(A) = sin(A) / cos(A) tan(A) = (40/41) / (9/41) tan(A) = 40/9 tan(B) = sin(B) / cos(B) tan(B) = (21/29) / (20/29) tan(B) = 21/20 Now we can find tan(A-B). tan(A-B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B)) tan(A-B) = ((40/9) - (21/20)) / (1 + (40/9)(21/20))Simplify the expression for tan(A-B). tan(A-B) = ((40/9) - (21/20)) / (1 + (40/9)(21/20)) To subtract the fractions, find a common denominator. Common denominator for 9 and 20 is 180. tan(A-B) = ((40*20)/(9*20) - (21*9)/(20*9)) / (1 + (40*21)/(9*20)) tan(A-B) = ((800/180) - (189/180)) / (1 + (840/180)) tan(A-B) = (800/180 - 189/180) / (1 + 840/180) tan(A-B) = (611/180) / (1020/180 + 180/180) tan(A-B) = (611/180) / (1200/180) Now simplify the expression. tan(A-B) = (611/180) * (180/1200) tan(A-B) = 611/1200

If $\sin A=\frac{40}{41}$ and $\cos B=\frac{20}{29}$ and angles A and B are in Quadrant I, find the value of $\tan (A-B)$ . $\newline$ Answer:

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If $\sin A=\frac{40}{41}$ and $\cos B=\frac{20}{29}$ and angles A and B are in Quadrant I, find the value of $\tan (A-B)$ . $\newline$ Answer: