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

Ask by French Young. in South Africa
Mar 11,2025

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Answer

\[ \sec A + \sin A = \frac{229}{156} \]

Solution

We start with the given equation: \[ 13\cos A = 12 \] which implies: \[ \cos A = \frac{12}{13}. \] Since \(\sin^2 A + \cos^2 A = 1\), we can find \(\sin A\) by: \[ \sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169}. \] Assuming \(A\) is an acute angle (so that \(\sin A\) is positive), we have: \[ \sin A = \frac{5}{13}. \] Now, we recall that the secant function is the reciprocal of the cosine function: \[ \sec A = \frac{1}{\cos A} = \frac{13}{12}. \] We are asked to find: \[ \sec A + \sin A = \frac{13}{12} + \frac{5}{13}. \] To add these fractions, we use a common denominator, which is \(12 \times 13 = 156\): \[ \frac{13}{12} = \frac{13 \times 13}{12 \times 13} = \frac{169}{156}, \] \[ \frac{5}{13} = \frac{5 \times 12}{13 \times 12} = \frac{60}{156}. \] Thus, the sum is: \[ \sec A + \sin A = \frac{169}{156} + \frac{60}{156} = \frac{229}{156}. \]

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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 \( \frac{229}{156} \).

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