Problem

(a)

The diagram shows the curve $y = k\cos\big(x - \frac{1}{6}\pi\big)$ where $k$ is a positive constant and $x$ is measured in radians. The curve crosses the $x$-axis at point $A$ and $B$ is a minimum point.

Find the coordinates of $A$ and $B$.

(b) Find the exact value of $t$ that satisfies the equation

$3\sin^{-1}(3t) + 2\cos^{-1}\bigg(\frac{1}{2}\sqrt{2}\bigg) = \pi.$