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If tan A=(24)/(7) and cos B=(12)/(37) and angles A and B are in Quadrant I, find the value of tan(A-B). Answer:

If tan⁡A=247 \tan A=\frac{24}{7} and cos⁡B=1237 \cos B=\frac{12}{37} and angles A and B are in Quadrant I, find the value of tan⁡(A−B) \tan (A-B) .\newlineAnswer:

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Q. If tan⁡A=247 \tan A=\frac{24}{7} and cos⁡B=1237 \cos B=\frac{12}{37} and angles A and B are in Quadrant I, find the value of tan⁡(A−B) \tan (A-B) .\newlineAnswer:
  1. Use tan⁡(A−B)\tan(A-B) formula: Use the formula for tan⁡(A−B)\tan(A-B), which is tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡A⋅tan⁡B\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B}. We know tan⁡A=247\tan A = \frac{24}{7}, but we need to find tan⁡B\tan B.
  2. Find sin⁡B\sin B: To find tan⁡B\tan B, we need to use the Pythagorean identity sin⁡2B+cos⁡2B=1\sin^2 B + \cos^2 B = 1 to find sin⁡B\sin B. We know cos⁡B=1237\cos B = \frac{12}{37}, so we can calculate sin⁡B\sin B. sin⁡2B=1−cos⁡2B=1−(1237)2\sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{12}{37}\right)^2.
  3. Calculate sin⁡2B\sin^2 B: Calculate sin⁡2B\sin^2 B.sin⁡2B=1−(1441369)=(13691369)−(1441369)=12251369\sin^2 B = 1 - (\frac{144}{1369}) = (\frac{1369}{1369}) - (\frac{144}{1369}) = \frac{1225}{1369}.
  4. Find tan⁡B\tan B: Find sin⁡B\sin B by taking the square root of sin⁡2B\sin^2 B.\newlineSince BB is in Quadrant I, sin⁡B\sin B is positive.\newlinesin⁡B=12251369=3537\sin B = \sqrt{\frac{1225}{1369}} = \frac{35}{37}.
  5. Simplify tan⁡B\tan B: Now we can find tan⁡B\tan B using sin⁡B\sin B and cos⁡B\cos B.\newlinetan⁡B=sin⁡Bcos⁡B=35371237\tan B = \frac{\sin B}{\cos B} = \frac{\frac{35}{37}}{\frac{12}{37}}.
  6. Apply tan⁡(A−B)\tan(A-B) formula: Simplify tan⁡B\tan B.tan⁡B=(3537)×(3712)=3512.\tan B = \left(\frac{35}{37}\right) \times \left(\frac{37}{12}\right) = \frac{35}{12}.
  7. Find common denominator: Now we have tan⁡A\tan A and tan⁡B\tan B, so we can use the tan⁡(A−B)\tan(A-B) formula.tan⁡(A−B)=tan⁡A−tan⁡B1+tan⁡A⋅tan⁡B=(247)−(3512)1+(247)⋅(3512)\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \cdot \tan B} = \frac{(\frac{24}{7}) - (\frac{35}{12})}{1 + (\frac{24}{7}) \cdot (\frac{35}{12})}.
  8. Perform subtraction: Find a common denominator and subtract 247\frac{24}{7} and 3512\frac{35}{12}. The common denominator is 7×12=847 \times 12 = 84. tan⁡(A−B)=(24×127×12−35×712×7)1+(24×357×12).\tan(A-B) = \frac{\left(\frac{24 \times 12}{7 \times 12} - \frac{35 \times 7}{12 \times 7}\right)}{1 + \left(\frac{24 \times 35}{7 \times 12}\right)}.
  9. Simplify numerator and denominator: Perform the subtraction in the numerator. tan⁡(A−B)=(288−245)/841+(840/84)\tan(A-B) = \frac{(288 - 245) / 84}{1 + (840/84)}.
  10. Divide numerator by denominator: Simplify the numerator and the term inside the parentheses in the denominator. tan⁡(A−B)=4384/(1+10)\tan(A-B) = \frac{43}{84} / (1 + 10).
  11. Simplify fraction: Simplify the denominator. tan⁡(A−B)=4384/11\tan(A-B) = \frac{43}{84} / 11.
  12. Simplify fraction: Simplify the denominator. tan⁡(A−B)=4384/11\tan(A-B) = \frac{43}{84} / 11. Divide the numerator by the denominator. tan⁡(A−B)=4384×11=43924\tan(A-B) = \frac{43}{84 \times 11} = \frac{43}{924}.
  13. Simplify fraction: Simplify the denominator. tan⁡(A−B)=4384/11\tan(A-B) = \frac{43}{84} / 11. Divide the numerator by the denominator. tan⁡(A−B)=4384×11=43924\tan(A-B) = \frac{43}{84 \times 11} = \frac{43}{924}. Simplify the fraction. tan⁡(A−B)=43924\tan(A-B) = \frac{43}{924} can be simplified by dividing both numerator and denominator by 44. tan⁡(A−B)=43924=434×231=43924=14×21=184\tan(A-B) = \frac{43}{924} = \frac{43}{4 \times 231} = \frac{43}{924} = \frac{1}{4 \times 21} = \frac{1}{84}.

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