2025 TJC P1 Q4

In the triangle \(\Delta ABC\), \(AC = 1\), \(\angle BAC = \frac{\pi}{3}\) radians and \(\angle ABC = \left( \frac{\pi}{6} + \theta \right)\) radians.

1. Show that \(BC = \frac{\sqrt{3}}{\cos \theta + \sqrt{3} \sin \theta}\).[2]

2. Given that \(\theta\) is sufficiently small such that \(\theta^3\) and higher powers of \(\theta\) may be neglected, show that\n\[BC \approx \sqrt{3} (1 + a\theta + b\theta^2)\] where \(a\) and \(b\) are constants to be determined.

   [3]