c) If $$ 13 \cos A=12 $$, find the value of $$ \sec A+\sin A $$.

Question

UpStudy AI · Accepted Answer

We are given
$$
13\cos A = 12.
$$
Thus,
$$
\cos A = \frac{12}{13}.
$$

Using the Pythagorean identity, we have
$$
\sin^2 A + \cos^2 A = 1.
$$
Substitute $$\cos A$$:
$$
\sin^2 A + \left(\frac{12}{13}\right)^2 = 1.
$$
$$
\sin^2 A + \frac{144}{169} = 1.
$$
Solve for $$\sin^2 A$$:
$$
\sin^2 A = 1 - \frac{144}{169} = \frac{169 - 144}{169} = \frac{25}{169}.
$$
Taking the positive square root (assuming $$A$$ is in the first quadrant):
$$
\sin A = \frac{5}{13}.
$$

Recall the definition of secant:
$$
\sec A = \frac{1}{\cos A} = \frac{1}{\frac{12}{13}} = \frac{13}{12}.
$$

Now, compute $$\sec A + \sin A$$:
$$
\sec A + \sin A = \frac{13}{12} + \frac{5}{13}.
$$
Find a common denominator, which is $$12 \times 13 = 156$$:
$$
\frac{13}{12} = \frac{13 \times 13}{156} = \frac{169}{156}, \quad \frac{5}{13} = \frac{5 \times 12}{156} = \frac{60}{156}.
$$
Thus,
$$
\sec A + \sin A = \frac{169}{156} + \frac{60}{156} = \frac{229}{156}.
$$

The final answer is 
$$
\frac{229}{156}.
$$
$$ \sec A + \sin A = \frac{229}{156}. $$

c) If , find the value of .

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