Since \(\theta\) is an obtuse angle, \(\theta\) is in the second quadrant where sine is positive and cosine is negative.

(i) Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we have:

\(\cos^2 \theta = 1 - \sin^2 \theta = 1 - k^2\)

\(\cos \theta = -\sqrt{1-k^2}\) because cosine is negative in the second quadrant.

(ii) \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{k}{-\sqrt{1-k^2}} = -\frac{k}{\sqrt{1-k^2}}\)

(iii) \(\sin(\theta + \pi) = -\sin \theta = -k\) because adding \(\pi\) to an angle reflects it across the origin, changing the sign of sine.