# Memorizing the unit circle?

I know a quick google brings up plenty of resources on memorization techniques for the unit circle but I thought I would get the math stack exchange's opinion.

• What is the best way to memorize the radian angles and their corresponding points on the unit circle? ( Right now I am thinking of just drawing it from scratch until I've got it.)

• Is memorizing the circle the best way to be able to quickly solve trig functions? (When first learning at least)

• How would I solve a trig function that takes an angle as an argument that has not been memorized? (You can memorize the angles that are common but there's no way you could memorize every possible angle between 0 and 2pi, is there?)

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Do you have an end goal, other than committing to memory tables of data? – Ryan Budney Apr 9 '11 at 1:16
@Ryan: Why would you need to do anything but memorize? That's all you get tested on. – Alex Becker Apr 9 '11 at 1:30
@Alex: I'm not sure I understand where you or Matt are coming from. It sounds like your concern then is doing well on some kind of standardized test? – Ryan Budney Apr 9 '11 at 1:43
@Ryan: At least at my high school, memorization of the unit circle is tested in precalculus. – Alex Becker Apr 9 '11 at 1:54
Even past highschool and pre-calculus, having memorized at least $\sin(\theta)$ and $\cos(\theta)$ for multiples of $\theta = \pi/4$ and $\theta = \pi/3$ has been implicitly required in many of the courses I've been in and TA'd. I've seen professors deduct a point for failing to evaluate $\sin(3\pi /4)$ for example, and in large classes with multiple choice tests, the available answers may only have the evaluated results. – matt Apr 9 '11 at 2:51

1. Varies widely.
2. Varies widely.
3. Depends. Any time you would encounter a trig function with very unusual arguments, it probably wouldn't be in a school exam so you'd simply use a calculator/computer. For simpler arguments which you haven't memorized, you can take advantage of half-angle, double-angle, and sum identities. For example: $\text{cos}(\frac{11\pi}{12}) = \text{cos}(\frac{2\pi}{3} + \frac{\pi}{4}) = \text{cos}(\frac{2\pi}{3})\text{cos}(\frac{\pi}{4}) - \text{sin}(\frac{2\pi}{3})\text{sin}(\frac{\pi}{4})$. These identities and much more, including memorization techniques for the unit circle, can be found here.
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The OP acknowledges the plethora of memorization techniques in his first line. – picakhu Apr 9 '11 at 1:34
@picakhu: I thought it might be worth mentioning that the wikipedia link also includes memorization techniques just in case one proves helpful. – Alex Becker Apr 9 '11 at 1:37
where in the Wikipedia article are memory techniques mentioned? thanks – Matt Apr 9 '11 at 1:57
@Matt: en.wikipedia.org/wiki/… – Alex Becker Apr 9 '11 at 2:01

I don't know about the fastest or "best" way, but I never bothered memorizing the unit circle. Rather, I memorized the pictures of the sine wave and cosine waves. (Maybe I should embed a picture here...)

Once you have the images of those curves in mind, together with the knowledge of the sine and cosine values of 0, 30, 45, 60, and 90 degrees, symmetry will take care of the rest.

For instance, if I wanted to compute $\sin 120$, I notice that it's 30 degrees from 90 degrees, so that by the symmetry of the sine wave, $\sin 120 = \sin 60 = \frac{\sqrt{3}}{2}$.

Anyway, memorizing the sine and cosine values of 0, 30, 45, 60, and 90 degrees shouldn't be too much of an issue anyway: for sine, it's just $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$ -- which I always remember as $$\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$$ -- and for cosine it's the reverse.

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and since $\tan = \frac{\sin}{\cos}$, for $\tan$ it is $\frac{\sqrt{0}}{\sqrt{4}},\frac{\sqrt{1}}{\sqrt{3}}, \frac{\sqrt{2}}{\sqrt{2}}, \frac{\sqrt{3}}{\sqrt{1}},\frac{\sqrt{4}}{\sqrt{0}}$ or so – Henry Apr 9 '11 at 17:53
1. Best is subjective.
2. Not necessarily, I found it easier to use 30,45,60 degree right triangles.
3. You cannot in general without a calculator, and you would not be expected to in general.
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You probably mean 45 degrees, not 40... ;) – Hans Lundmark Apr 9 '11 at 6:44
All you really need to remember is that $1,1,\sqrt{2}$ gives a triangle which is half a square and $1,\sqrt{3},2$ gives a triangle which is half an equilateral triangle, plus perhaps the shape of a sine curve repeating every $2\pi$ starting at $\sin{0}=0$, and a cosine curve being shifted along so $\cos{0}=1$. – Henry Apr 9 '11 at 17:50
or the Pythagoras theorem – picakhu Apr 9 '11 at 18:00

Check out this Youtube video about the Unit-circle (A way to remember the Entire Unit Circle for Trigonometry) http://youtu.be/cIVpemcoAlY

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