Start with the equation \(\tan \theta \sin \theta = 1\).
Using \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we have:
\(\frac{\sin^2 \theta}{\cos \theta} = 1\).
Multiply both sides by \(\cos \theta\):
\(\sin^2 \theta = \cos \theta\).
Using the identity \(\sin^2 \theta = 1 - \cos^2 \theta\), substitute:
\(1 - \cos^2 \theta = \cos \theta\).
Rearrange to form a quadratic equation:
\(\cos^2 \theta + \cos \theta - 1 = 0\).
Use the quadratic formula \(\cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1, b = 1, c = -1\):
\(\cos \theta = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}\).
\(\cos \theta = \frac{-1 \pm \sqrt{5}}{2}\).
Calculate the possible values for \(\cos \theta\):
\(\cos \theta = \frac{-1 + \sqrt{5}}{2} \approx 0.618\).
Find \(\theta\) using \(\cos^{-1}(0.618)\):
\(\theta \approx 51.8^\circ\) and \(\theta \approx 360^\circ - 51.8^\circ = 308.2^\circ\).
