If $\tan A=\frac{4}{3}$ and $\cos B=\frac{8}{17}$ and angles A and B are in Quadrant I, find the value of $\tan (A+B)$ . $\newline$ Answer:

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Q. If

\tan A=\frac{4}{3}

and

\cos B=\frac{8}{17}

and angles A and B are in Quadrant I, find the value of

\tan (A+B)

\newline

Answer:

Apply Tangent Addition Formula: Use the tangent addition formula: $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}$ . First, we need to find $\tan B$ . Since $\cos B = \frac{8}{17}$ and $B$ is in Quadrant I, where all trigonometric functions are positive, we can find $\sin B$ using the Pythagorean identity $\sin^2 B + \cos^2 B = 1$ .
Calculate sin B: Calculate sin B: $\sin^2 B = 1 - \cos^2 B = 1 - \left(\frac{8}{17}\right)^2 = 1 - \frac{64}{289} = \frac{289 - 64}{289} = \frac{225}{289}$ . Therefore, $\sin B = \sqrt{\frac{225}{289}} = \frac{15}{17}$ , since we are in Quadrant I and sine is positive.
Find $\tan B$ : Now, find $\tan B$ using the quotient of $\sin B$ and $\cos B$ : $\tan B = \frac{\sin B}{\cos B} = \frac{\frac{15}{17}}{\frac{8}{17}} = \frac{15}{8}$ .
Substitute tan values: Substitute the values of $\tan A$ and $\tan B$ into the tangent addition formula: $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B} = \frac{\left(\frac{4}{3}\right) + \left(\frac{15}{8}\right)}{1 - \left(\frac{4}{3}\right) \cdot \left(\frac{15}{8}\right)}$ .
Simplify numerator: Simplify the numerator: $(\frac{4}{3}) + (\frac{15}{8}) = (\frac{32}{24}) + (\frac{45}{24}) = (\frac{32 + 45}{24}) = \frac{77}{24}$ . Simplify the denominator: $1 - (\frac{4}{3}) \times (\frac{15}{8}) = 1 - (\frac{60}{24}) = (\frac{24}{24}) - (\frac{60}{24}) = \frac{-36}{24} = \frac{-3}{2}$ .
Simplify denominator: Now, divide the numerator by the denominator: $\tan(A+B) = \frac{77}{24} / \left(-\frac{3}{2}\right) = \frac{77}{24} \cdot \left(-\frac{2}{3}\right) = -\frac{154}{72}.$
Divide numerator by denominator: Simplify the fraction $-154 / 72$ by dividing both numerator and denominator by their greatest common divisor, which is $2$ : $\tan(A+B) = (-154 / 2) / (72 / 2) = -77 / 36$ .