If 5sinθ - 12cosθ = 0

find the value of \(\frac{1+sinθ+cosθ}{1-sinθ+cosθ}\).

\(\frac{3}{2}\)

5sinθ - 12cosθ = 0

5sinθ = 12cosθ 

\(\frac{sinθ}{cosθ}\) = \(\frac{12}{5}\)

tanθ = \(\frac{12}{5}\) = \(\frac{P}{B}\)  [Triplet 5, 12, 13]

P = 12, B = 5, H = 13,

⇒ sinθ = \(\frac{P}{H}\) = \(\frac{12}{13}\), and

⇒ cosθ = \(\frac{B}{H}\) = \(\frac{5}{13}\)

Now,

⇒ \(\frac{1\;+\;sinθ\;+\;cosθ}{1\;-\;sinθ\;+\;cosθ}\) = \(\frac{1\;+\;\frac{12}{13}\;+\;\frac{5}{13}}{1\;-\;\frac{12}{13}\;+\;\frac{5}{13}}\) = \(\frac{30}{20}\) = \(\frac{3}{2}\)