Trig Functions of Angles Lying on the Axes

At this point, we need to stop and talk about 4 special angles, which are the ones whose terminal sides lie on the axes. Namely, if is in standard position, then the angles in question are 0°, 90°, 180°, and 270°. In radians, these angles are 0, , , and radians.

Notice that one cannot build an appropriate right triangle for these 4 angles. This seems to cause problems for finding the trig function values at these angles. However, we can easily remedy this.

Note that, in the past, we built the various right triangles in a manner given in the figure below:

Then, we see that the trig functions can actually be defined as the following:

Note that these are exactly the same definitions as were given before, except that we are now referring to x, y, and r rather than the adjacent leg, the opposite leg, and the hypotenuse.

Given this focus now on x, y, and r, we can now find the trigonometric values at the angles 0°, 90°, 180°, and 270°. We begin with the angle .

Note in the case that we have x=0, y=1, and r=1. See the diagram below.

(Realize that we could let y=2 or 3 or any positive number for that matter. If we let y=2 for example, then we will have r=2 also.) Then, we can find the values of the six trig functions quickly using the definitions mentioned above in terms of x, y, and r. We have

Similarly, in the case of , we have x=-1, y=0, and r=1. (We will always think of r as the distance from the point (x,y) to the origin (0,0). In that sense then, r will always be nonnegative.) These values then yield the following:

One can easily calculate the values of the trig functions for and . We leave this to the reader.

To summarize the results of this section, we close with the following table:

0°	0	0	1	0	und	1	und
90°		1	0	und	1	und	0
180°		0	-1	0	und	-1	und
270°		-1	0	und	-1	und	0