If sin A 513 for angle A in Quadrant I find cos 2A?

To find ( \cos 2A ) using the given ( \sin A = \frac{5}{13} ), we first use the Pythagorean identity to find ( \cos A ). Since ( \sin^2 A + \cos^2 A = 1 ), we have ( \cos^2 A = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169} ). Thus, ( \cos A = \frac{12}{13} ). Using the double angle formula ( \cos 2A = 2\cos^2 A - 1 ), we get ( \cos 2A = 2\left(\frac{12}{13}\right)^2 - 1 = 2 \cdot \frac{144}{169} - 1 = \frac{288}{169} - \frac{169}{169} = \frac{119}{169} ).