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

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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.

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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.

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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.

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

google image search for pi/6

But it's also good to develop "(2) skill of understanding" by being able to quickly come up with the answer, like using flashcards, to make the understanding instinctual and fast. And of course, (3) applying the knowledge by solving problems. So you're neither "just memorizing facts" nor "just understanding and not doing the homework". It's the combination of (1) understanding, (2) "skill of understanding", and (3) problem solving that develops a good foundation for the mathematical mind.

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?

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