\( f 13 \cos A=12 \), find the value of \( \sec A+\sin \)

To find the value of \( \sec A + \sin A \) given the equation \( f 13 \cos A = 12 \), we first need to clarify the equation. Assuming the intention was to say \( 13 \cos A = 12 \): 1. We can isolate \( \cos A \): \[ \cos A = \frac{12}{13}. \] 2. Next, we can find \( \sin A \) using the Pythagorean identity: \[ \sin^2 A + \cos^2 A = 1. \] Substituting \( \cos A \): \[ \sin^2 A + \left(\frac{12}{13}\right)^2 = 1 \implies \sin^2 A + \frac{144}{169} = 1. \] This simplifies to: \[ \sin^2 A = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169}. \] Therefore: \[ \sin A = \frac{5}{13} \quad (\text{since } A \text{ is in the first quadrant, } \sin A \text{ is positive}). \] 3. Now, we can find \( \sec A \): \[ \sec A = \frac{1}{\cos A} = \frac{1}{\frac{12}{13}} = \frac{13}{12}. \] 4. Finally, we can calculate \( \sec A + \sin A \): \[ \sec A + \sin A = \frac{13}{12} + \frac{5}{13}. \] To add these fractions, we find a common denominator (which is 156): \[ \sec A + \sin A = \frac{13 \times 13}{156} + \frac{5 \times 12}{156} = \frac{169 + 60}{156} = \frac{229}{156}. \] Thus, the value of \( \sec A + \sin A \) is \( \frac{229}{156} \).