The range of values of θ in the interval (0, π) such that the points (3,5) and (sinθ, cosθ) lie on the same side of the line x + y − 1 = 0, is

Same Side of a Line — Geometric Condition

Line: \( x + y - 1 = 0 \)
Point 1: (3, 2) → Lies on the side where value is positive:
\[ f(3, 2) = 3 + 2 - 1 = 4 > 0 \]

Point 2: \( (\cos\theta, \sin\theta) \) lies on same side if: \[ \cos\theta + \sin\theta > 1 \] Using identity: \[ \cos\theta + \sin\theta = \sqrt{2} \sin\left(\theta + \frac{\pi}{4}\right) \Rightarrow \sin\left(\theta + \frac{\pi}{4}\right) > \frac{1}{\sqrt{2}} \] So: \[ \theta \in \left(0,\ \frac{\pi}{2}\right) \]

✅ Final Answer: \( \boxed{\theta \in \left(0,\ \frac{\pi}{2}\right)} \)