If $\sin A = \frac{5}{13}$, then find the value of

$$\frac{\cos A - 2 an A}{\sin A + 3 an A}$$

Question

If $\sin A = \frac{5}{13}$, then find the value of

$$\frac{\cos A - 2 	an A}{\sin A + 3 	an A}$$

Accepted Answer

**Solution:**

1. Given $\sin A = \frac{5}{13}$.

2. Use the Pythagorean identity:
   $$\cos^2 A + \sin^2 A = 1$$
   $$\Rightarrow \cos^2 A = 1 - \sin^2 A = 1 - \left(\frac{5}{13}ight)^2 = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169}$$

3. Therefore,
   $$\cos A = \frac{12}{13}$$ (taking positive value assuming angle in first quadrant).

4. Calculate $	an A$:
   $$	an A = \frac{\sin A}{\cos A} = \frac{\frac{5}{13}}{\frac{12}{13}} = \frac{5}{12}$$

5. Substitute values into the expression:
   $$\frac{\cos A - 2 	an A}{\sin A + 3 	an A} = \frac{\frac{12}{13} - 2 	imes \frac{5}{12}}{\frac{5}{13} + 3 	imes \frac{5}{12}}$$

6. Calculate numerator:
   $$\frac{12}{13} - \frac{10}{12} = \frac{12 	imes 12}{13 	imes 12} - \frac{10 	imes 13}{12 	imes 13} = \frac{144}{156} - \frac{130}{156} = \frac{14}{156} = \frac{7}{78}$$

7. Calculate denominator:
   $$\frac{5}{13} + \frac{15}{12} = \frac{5 	imes 12}{13 	imes 12} + \frac{15 	imes 13}{12 	imes 13} = \frac{60}{156} + \frac{195}{156} = \frac{255}{156} = \frac{85}{52}$$

8. Now the expression becomes:
   $$\frac{7/78}{85/52} = \frac{7}{78} 	imes \frac{52}{85} = \frac{7 	imes 52}{78 	imes 85}$$

9. Simplify numerator and denominator:
   - $7 	imes 52 = 364$
   - $78 	imes 85 = 6630$

10. Simplify fraction:
    $$\frac{364}{6630} = \frac{14}{255}$$ (dividing numerator and denominator by 26)

**Answer:** $\frac{14}{255}$

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