Answer: B
\(5\cos\theta + 12\sin\theta = 13\) Or, \(5 + 12\cfrac{\sin\theta}{\cos\theta} = \cfrac{13}{\cos\theta}\) Or, \(5 + 12\tan\theta = 13\sec\theta\) Or, \((5 + 12\tan\theta)^2 = (13\sec\theta)^2\) Or, \(25 + 2 \cdot 5 \cdot 12\tan\theta + 144\tan^2\theta = 169\sec^2\theta\) Or, \(25 + 120\tan\theta + 144\tan^2\theta = 169(1 + \tan^2\theta)\) Or, \(25 + 120\tan\theta + 144\tan^2\theta - 169 - 169\tan^2\theta = 0\) Or, \(-25\tan^2\theta + 120\tan\theta - 144 = 0\) Or, \(25\tan^2\theta - 120\tan\theta + 144 = 0\) Or, \((5\tan\theta)^2 - 2 \cdot 5\tan\theta \cdot 12 + 12^2 = 0\) Or, \((5\tan\theta - 12)^2 = 0\) Or, \(5\tan\theta - 12 = 0\) Or, \(5\tan\theta = 12\) Or, \(\tan\theta = \cfrac{12}{5}\)
\(5\cos\theta + 12\sin\theta = 13\) Or, \(5 + 12\cfrac{\sin\theta}{\cos\theta} = \cfrac{13}{\cos\theta}\) Or, \(5 + 12\tan\theta = 13\sec\theta\) Or, \((5 + 12\tan\theta)^2 = (13\sec\theta)^2\) Or, \(25 + 2 \cdot 5 \cdot 12\tan\theta + 144\tan^2\theta = 169\sec^2\theta\) Or, \(25 + 120\tan\theta + 144\tan^2\theta = 169(1 + \tan^2\theta)\) Or, \(25 + 120\tan\theta + 144\tan^2\theta - 169 - 169\tan^2\theta = 0\) Or, \(-25\tan^2\theta + 120\tan\theta - 144 = 0\) Or, \(25\tan^2\theta - 120\tan\theta + 144 = 0\) Or, \((5\tan\theta)^2 - 2 \cdot 5\tan\theta \cdot 12 + 12^2 = 0\) Or, \((5\tan\theta - 12)^2 = 0\) Or, \(5\tan\theta - 12 = 0\) Or, \(5\tan\theta = 12\) Or, \(\tan\theta = \cfrac{12}{5}\)