Generalizing other Trigonometric Functions

In a manner consistent with right-triangle trigonometry, we define generalized versions of the tangent, secant, cotangent, and cosecant functions in terms of the sine and cosine functions as follows:

$$\begin{array}{l} \left. \begin{array}{rcl} \tan t &=& \frac{\sin t}{\cos t}\\\\ \sec t &=& \frac{1}{\cos t} \end{array} \quad \right\} \quad \overset{\normalsize{\textrm{for all } t \textrm{ except } \frac{\pi}{2} \pm n\pi, \,\, n = 0,1,2,\;\ldots}}{\scriptsize{\textrm{(odd multiples of } \pi/2 \textrm{)}}}\\\\ \left. \begin{array}{rcl} \cot t &=& \frac{\cos t}{\sin t}\\\\ \csc t &=& \frac{1}{\sin t} \end{array} \quad \right\} \quad \overset{\normalsize{\textrm{for all } t \textrm{ except } n\pi, \,\, n = 0,1,2,\;\ldots}}{\scriptsize{\textrm{(integer multiples of } \pi \textrm{)}}} \end{array}$$

Note the restrictions on the domains of these other trigonometric functions. The values of $t$ excluded from their domains are those for which the denominators are equal to zero.

Since the other trigonometric functions are defined in terms of the sine and cosine, values of these other functions can easily be obtained for any $t$ for which the sine and cosine are known.