Clearly, the point $\left(-5,12\right)$ lies in second quadrant. Since, it's $x-$coordinate is negative and $y-$coordinate is positive.

We know that, in second quadrant $\sin \theta \& \cos ec\theta$ are positive and $\cos \theta , sec\theta , \tan \theta \& cot\theta$ are negative.

So, by using definition of modulus function, we get $\left|\sin \theta \right|=\sin \theta$, $\left|\cos \theta \right|=-\cos \theta$, $\left|\tan \theta \right|=-\tan \theta$, $\left|\cos ec\theta \right|=\cos ec\theta$.