Q: Let P (sinθ , cosθ) (0 ≤ θ ≤ 2π) be a point and let OAB be a triangle with vertices (0, 0), (√(3/2),0 ) and (0 ,√(3/2) ) . Find θ if P lies inside the ΔOAB.

Sol. Equations of lines along OA, OB and AB are y = 0, x = 0, and $ x + y = \sqrt{\frac{3}{2}}$ respectively.
Now P and B will lie on the same side of y = 0 if cosθ > 0.

Similarly P and A will lie on the same side of x = 0 if sin θ > 0 and P and O will lie on the same side of $ x + y = \sqrt{\frac{3}{2}}$ if $ sin\theta + cos\theta < \sqrt{\frac{3}{2}} $ .

Hence P will lie inside the ΔABC, if sinθ > 0, cosθ > 0 and $ sin\theta + cos\theta < \sqrt{\frac{3}{2}} $

Now $ sin\theta + cos\theta < \sqrt{\frac{3}{2}} $