Axis Values
Recall that given an $(x, y)$ coordinate on the circumference of the unit circle, by definition $x = \cos \theta$ and $y = \sin \theta$. From this, the values of the trigonometric functions on the axes are essentially given.
From the above circle, we can obtain the values for $\sin$ and $\cos$. Then we can use the reciprocal and ratio identities, we can derive the rest. I won’t bother doing all the tedious proofs since they are very easy. The results are summarized below.
| Degrees | Radians | cos | sin | tan | sec | csc | cot |
|---|---|---|---|---|---|---|---|
| $0^{\circ}$ | $0$ | $1$ | $0$ | $0$ | $1$ | undefined | undefined |
| $90^{\circ}$ | $\pi/2$ | $0$ | $1$ | undefined | undefined | $1$ | $0$ |
| $180^{\circ}$ | $\pi$ | $-1$ | $0$ | $0$ | $1$ | undefined | undefined |
| $270^{\circ}$ | $3\pi/2$ | $0$ | $-1$ | undefined | undefined | $1$ | $0$ |
The undefined values are the result of division by zero. Depending on the context, both $+\infty$ or $-\infty$ could be reasonably assigned to these values, thus they have to remain undefined. Look back at the graphs of the trig functions in the Intuition of Trigonometric Functions post, and notice that the asymptotes line up with the undefined values.