If $\cos A=\frac{35}{37}$ and $\sin B=\frac{28}{53}$ and angles A and B are in Quadrant I, find the value of $\tan (A+B)$ . $\newline$ Answer:

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Find trigonometric ratios using multiple identities

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Q. If

\cos A=\frac{35}{37}

and

\sin B=\frac{28}{53}

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

\tan (A+B)

\newline

Answer:

Find $\sin A$ and $\cos B$ : Use the given values to find $\sin A$ and $\cos B$ . Since $\cos A = \frac{35}{37}$ , we can use the Pythagorean identity $\sin^2 A + \cos^2 A = 1$ to find $\sin A$ . $\sin^2 A = 1 - \cos^2 A$ $\sin^2 A = 1 - \left(\frac{35}{37}\right)^2$ $\sin^2 A = 1 - \frac{1225}{1369}$ $\cos B$ $0$ $\cos B$ $1$ $\cos B$ $2$ $\cos B$ $3$
Find $\cos B$ using $\sin B$ : Similarly, find $\cos B$ using $\sin B = \frac{28}{53}$ .
$\cos^2 B = 1 - \sin^2 B$
$\cos^2 B = 1 - \left(\frac{28}{53}\right)^2$
$\cos^2 B = 1 - \frac{784}{2809}$
$\cos^2 B = \frac{2809}{2809} - \frac{784}{2809}$
$\cos^2 B = \frac{2025}{2809}$
$\cos B = \sqrt{\frac{2025}{2809}}$
$\sin B$ $0$
Use angle sum identity for tangent: Use the angle sum identity for tangent to find $\tan(A+B)$ . $\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}$ Since $\tan A = \frac{\sin A}{\cos A}$ and $\tan B = \frac{\sin B}{\cos B}$ , we can substitute the values we found. $\tan A = \frac{12/37}{35/37} = \frac{12}{35}$ $\tan B = \frac{28/53}{45/53} = \frac{28}{45}$ $\tan(A+B) = \frac{\frac{12}{35} + \frac{28}{45}}{1 - \left(\frac{12}{35} \cdot \frac{28}{45}\right)}$
Simplify expression for $\tan(A+B)$ : Simplify the expression for $\tan(A+B)$ . $\tan(A+B) = \frac{12}{35} + \frac{28}{45}$ / $1 - \left(\frac{12}{35} \cdot \frac{28}{45}\right)$ To add the fractions, find a common denominator, which is $35\cdot45$ . $\tan(A+B) = \frac{(12\cdot45) + (28\cdot35)}{35\cdot45 - (12\cdot28)}$ $\tan(A+B) = \frac{540 + 980}{1575 - 336}$ $\tan(A+B) = \frac{1520}{1239}$
Check for simplification: Check for any possible simplification of the fraction. $\newline$ The numerator and denominator of $\frac{1520}{1239}$ do not have any common factors other than $1$ , so the fraction is already in its simplest form.