Answer
$\displaystyle \sin t=-\frac{24}{25}\quad \tan t=\frac{24}{7}$
$\displaystyle \sec t=-\frac{25}{7}\quad\csc t=-\frac{25}{24}\quad\cot t=\frac{7}{24}$
Work Step by Step
$\sin^{2}t+\cos^{2}t=1\qquad\Rightarrow\sin t=\pm\sqrt{1-\cos^{2}t}$
In quadrant $III$, tan and cot are positive,
all the other functions are negative (see table of signs, p.410)
So
$\sin t=-\sqrt{1-(-\frac{7}{25})^{2}}=-\sqrt{\dfrac{625-49}{625}}$
$=-\displaystyle \sqrt{\frac{576}{625}}=-\frac{24}{25}$
$\displaystyle \tan t=\frac{\sin t}{\cos t}=\frac{-\frac{24}{25}}{-\frac{7}{25}}=\frac{24}{7}$
$\displaystyle \sec t=\frac{1}{\cos t}=-\frac{25}{7}$ $\\$
$\displaystyle \csc t=\frac{1}{\sin t}=-\frac{25}{24}$ $\\$
$\displaystyle \cot t=\frac{1}{\tan t}=\frac{7}{24}$
