# How to memorize the families that are $\sin$, $\cos$, and $\tan$ of $\pi$ over something?

Is there any way to easily memorize these? Such as $\sin \frac{\pi}{6} = 1/2$. Any help needed!!!

Here's a nice trick:

$$\sin\left(0\right) = \sqrt{\frac{0}{4}}\\ \sin\left(\frac{\pi}{6}\right) = \sqrt{\frac{1}{4}}\\ \sin\left(\frac{\pi}{4}\right) = \sqrt{\frac{2}{4}}\\ \sin\left(\frac{\pi}{3}\right) = \sqrt{\frac{3}{4}}\\ \sin\left(\frac{\pi}{2}\right) = \sqrt{\frac{4}{4}}$$

where the angles are in increasing order.

• This is pretty cool. :) – tomasz Jan 6 '14 at 1:32
• The same sort of trick works for $\cos$, but angles in decreasing order (in case that wasn't immediately clear for anyone). – apnorton Jan 6 '14 at 1:55

I assume you mean $\pi/4$, $\pi/3$, $\pi/6$ and their multiples.

For sine and cosine, either draw a right triangle with the corresponding angle or draw a rough plot, remembering that there's a $\frac{1}{2}$ and $\frac{\sqrt 3}{2}$ somewhere. For tangent and cotangent just calculate the quotient.

Some others are easily calculated using trigonometric identities.

I just remember that a 45-45-90 triangle is a square cut in half, and a 30-60-90 triangle is an equilateral triangle cut in half. Using the theorem of Pythagoras (which I memorized a long time ago) I can work out the side lengths. I also have to remember which is the sine and which is the cosine. My mnemonic for that is that for $(x,y)$ on the unit circle, $(x,y)=(\cos\theta,\sin\theta)$ where both sides of the equation are in alphabetical order. I find that a lot easier than using a vaguely racist mnemonic about an Indian Chief whose name I can never remember. OK, I think it's "Sohcahtoa" but my mnemonic for remembering "Sohcahtoa" is the trigonometric ratios.

(1) Visualize it so you understand it (and don't have to memorize it):

FYI, here's another good example of "visualizing for memorizing": Is there any intuition behind why the derivative of $\cos(x)$ is equal to $-\sin(x)$ or just something to memorize?