Using fundamental identities to find the value of the trigonometric function, find the sin theta if cos = -square root of 11/6 and theta is in quadrant III

I assume you meant \cos \theta = -\sqrt{11/16} not -sqrt{11/6}.

Given:


\cos \theta = - \sqrt {11/16}


Use the fundamental identity


\sin^2 \theta + \cos^2 \theta = 1


Rearrange to get sin as a function of cosine:


\sin \theta = \sqrt {1 - \cos^2 \theta}


For this problem you know the value of \cos \theta , so plug and chug. Use the fact that \theta is in the 3rd quadrant to determine the correct sign for \sin \theta .