Reference Triangles, Reference Angles

Now we extend our ``right triangle'' definitions of the trig functions to have a domain of all real numbers, not just numbers in the interval $0<\theta<\pi/2.$

Consider again our right triangle in the first quadrant, with hypotenuse corresponding to the terminal side of $\theta$ and adjacent side corresponding to the $x$ -axis. There exists a similar triangle in QII, also with hypotenuse 1 and sharing a side with the $x$ -axis. There exists similar triangles in QIII and QIV also, all with hypotenuse 1, and all sharing a side with the $x$ -axis.

Because of the properties of similar triangles, each of these triangles has an acute angle $\alpha$ (always taken positive) equal to our original $\theta,$ and all six trigonometric ratios of any of these similar triangles are equal to the corresponding ratios of our original triangle in QI. For that matter, the hypotenuse of these triangles need not be 1 and the same holds true, again due to properties of similar triangles.

With respect to an angle $\theta$ in standard position, such a triangle is called a reference triangle (or related triangle.) The angle $\alpha$ (always taken positive) between the terminal side of $\theta$ and the $x$ -axis, is called a reference angle (or related angle.)

For example, when $\theta=2\pi/3$ (120 degrees), the reference angle is $\pi/3$ (60 degrees.)

Since the hypotenuse of such a triangle corresponds to the terminal side of an angle in standard position, every real number angle has such a reference triangle and reference angle (except for quadrantal angles which will be considered later.) We may use any reference triangle to consider a trigonometric ratio of a given angle, just as we do for the reference triangle in the first quadrant. When so doing, however, we have to assign the correct sign (plus or minus) to the ratio's value, as discussed previously, depending on which quadrant the terminal side of $\theta$ lies in. To continue with our example, since $2\pi/3$ has reference angle $\pi/3,\;\sin (2\pi/3)=\sin(\pi/3),$ while $\cos(2\pi/3)=-\cos(\pi/3),$ since sine is positive in both QI and QII, while cosine is positive in QI but negative in QII.

In closing our description of reference triangles, it is worth mentioning that although the hypotenuse of such a triangle need not be 1, it is usually convenient to let the hypotenuse be 1 when considering a reference triangle/angle, since this simplifies matters.