One weird trick! Click here!
Somewhere in my years of “tutoring calculus,” also sometimes referred to as “tutoring trigonometry,” I discovered a trick (here meaning a tool devoid of any conceptual knowledge of the topic at all) for evaluating the cosine (and therefore also sine) of the upper-right quadrant special angles . This can be extended to the rest of the unit circle in the usual way.
I rarely see this move discussed, and it is such an easy way to “memorize the unit circle” that I find this surprising. As with all academics, I should be writing, so here you go.
This post is for trigonometry students. At this point you know that an angle is a point on an origin-centered circle of radius 1, or the unit circle, and that cosine and sine are something that you do to those angles to get a number. You may also know that these numbers are the sides of the right triangle generated by the angle in question. What, exactly, cosine and sine describe is worth knowing, but I’d like to actually publish a complete post with a beginning and an end for once. Let’s get to the trick.
The special angles are points along the upper-right quadrant, or between and , that have “nice” values of cosine and sine. In other words, their values of cosine and sine may be represented in terms of fractions and square roots—things that are relatively easy to write down. These angles are
The cosine and sine values of each special angle are listed in the table below:
This is, on its own, perfectly readable. As a trigonometry student, you know that cosine measures the “horizontal part” of angle: in other words, how much of the angle is moving left-and-right as opposed to up-and-down? You know that the angle is completely located on the horizontal axis, and so that it is a 100% horizontal angle. Therefore . Moving counter-clockwise, our angle is less horizontal and more vertical, so its cosine value shrinks. Drawing triangles and using the Pythagorean theorem told you that the next cosine values are and respectively. Finally, the angle is located on the vertical axis, so it is a 0% horizontal angle. That’s why . (Sine goes the same way, but backwards.)
That’s conceptual knowledge of cosine and sine, which I promised you wouldn’t need. Here’s the trick. Observe that and that . Now here’s how our table looks:
Now cosine proceeds by just subtracting one from the thing in the square root! Sine, conversely, adds one inside the square root with each special angle.
As you’ve learned, in the upper-left quadrant, cosine values become negative; in the bottom-left quadrant, both cosine and sine are negative; and in the bottom-right quadrant, sine is negative. With this, you know have a tool that produces the cosine and sine values for special angles as quickly as you can count. I’d go so far as to say you aren’t even memorizing anything!
I have some questions about the trick that may beget a future post (not to belabor the point, but I really need to get something written and out). Here’s one: What if we defined a curve such that the and coordinates of the curve are determined by which special angle you are on without the unit-circular condition ? What would the polar expression of said curve look like? (In other words, can we reverse-engineer the circle from this trick with the special angles? Can we do it with other curves?) Here’s another: can this trick be explained in such a way that ties counting up and down inside the square root with the conceptual truth behind cosine and sine? If we define special angles to be angles whose cosine and sine values have the form , could we modify that definition to yield different special angles?
Anyway, there’s the trick. Trigonometry is hard and memorization is a poor substitute for learning: at least now you’re thinking about cosine and sine ebbing and flowing with your position on the circle!