TMUA practice: Trig Equations, problem 5

Write the line through $(1, 2)$ in parametric form

$(x, y) = (1 + t\cos\theta,\; 2 + t\sin\theta),$

where $|t|$ is the distance from $(1, 2)$ to the point with parameter $t$.

Imposing $P$ on $x + y = 4$:

For $0 < \theta \le \tfrac{\pi}{3}$ we have $\cos\theta + \sin\theta > 0$, so $t > 0$ and the distance equals $t = \dfrac{\sqrt6}{3}$. Hence

Solving the trig equation. Using the identity $\cos\theta + \sin\theta = \sqrt2\,\sin\!\left(\theta + \tfrac{\pi}{4}\right)$:

So $\theta + \dfrac{\pi}{4} = \dfrac{\pi}{3}$ or $\dfrac{2\pi}{3}$, giving $\theta = \dfrac{\pi}{12}$ or $\theta = \dfrac{5\pi}{12}$.

The constraint $0 < \theta \le \dfrac{\pi}{3} = \dfrac{4\pi}{12}$ admits $\dfrac{\pi}{12}$ and rejects $\dfrac{5\pi}{12}$.

Answer: B.