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If tan A=(4)/(3) and cos B=(8)/(17) and angles A and B are in Quadrant I, find the value of tan(A+B). Answer:

If tan⁡A=43 \tan A=\frac{4}{3} and cos⁡B=817 \cos B=\frac{8}{17} 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=43 \tan A=\frac{4}{3} and cos⁡B=817 \cos B=\frac{8}{17} and angles A and B are in Quadrant I, find the value of tan⁡(A+B) \tan (A+B) .\newlineAnswer:
  1. Apply Tangent Addition Formula: Use the tangent addition formula: 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}. First, we need to find tan⁡B\tan B. Since cos⁡B=817\cos B = \frac{8}{17} and BB is in Quadrant I, where all trigonometric functions are positive, we can find sin⁡B\sin B using the Pythagorean identity sin⁡2B+cos⁡2B=1\sin^2 B + \cos^2 B = 1.
  2. Calculate sin B: Calculate sin B: sin⁡2B=1−cos⁡2B=1−(817)2=1−64289=289−64289=225289\sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{8}{17}\right)^2 = 1 - \frac{64}{289} = \frac{289 - 64}{289} = \frac{225}{289}. Therefore, sin⁡B=225289=1517\sin B = \sqrt{\frac{225}{289}} = \frac{15}{17}, since we are in Quadrant I and sine is positive.
  3. Find tan⁡B\tan B: Now, find tan⁡B\tan B using the quotient of sin⁡B\sin B and cos⁡B\cos B: tan⁡B=sin⁡Bcos⁡B=1517817=158\tan B = \frac{\sin B}{\cos B} = \frac{\frac{15}{17}}{\frac{8}{17}} = \frac{15}{8}.
  4. Substitute tan values: Substitute the values of tan⁡A\tan A and tan⁡B\tan B into the tangent addition formula: tan⁡(A+B)=tan⁡A+tan⁡B1−tan⁡A⋅tan⁡B=(43)+(158)1−(43)⋅(158)\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} = \frac{\left(\frac{4}{3}\right) + \left(\frac{15}{8}\right)}{1 - \left(\frac{4}{3}\right) \cdot \left(\frac{15}{8}\right)}.
  5. Simplify numerator: Simplify the numerator: (43)+(158)=(3224)+(4524)=(32+4524)=7724(\frac{4}{3}) + (\frac{15}{8}) = (\frac{32}{24}) + (\frac{45}{24}) = (\frac{32 + 45}{24}) = \frac{77}{24}. Simplify the denominator: 1−(43)×(158)=1−(6024)=(2424)−(6024)=−3624=−321 - (\frac{4}{3}) \times (\frac{15}{8}) = 1 - (\frac{60}{24}) = (\frac{24}{24}) - (\frac{60}{24}) = \frac{-36}{24} = \frac{-3}{2}.
  6. Simplify denominator: Now, divide the numerator by the denominator: tan⁡(A+B)=7724/(−32)=7724⋅(−23)=−15472.\tan(A+B) = \frac{77}{24} / \left(-\frac{3}{2}\right) = \frac{77}{24} \cdot \left(-\frac{2}{3}\right) = -\frac{154}{72}.
  7. Divide numerator by denominator: Simplify the fraction −154/72-154 / 72 by dividing both numerator and denominator by their greatest common divisor, which is 22: tan⁡(A+B)=(−154/2)/(72/2)=−77/36\tan(A+B) = (-154 / 2) / (72 / 2) = -77 / 36.

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