When I was first introduced to trigonometric ratios, I learned the following trick to remember the trigonometric table of standard angles ($0^{\circ}$, $30^{\circ}$, $45^{\circ}$, $60^{\circ}$, $90^{\circ}$):
Sine: Take the numbers $0$, $1$, $2$, $3$, $4$ for each angle $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, $90^\circ$. Divide each number by $4$ and take the square root of the result. This will give the $\sin$ ratio of each corresponding angle.
Tangent: Take the numbers $0$, $1$, $3$, $9$ for each angle $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$. Divide each number by $3$ and take the square root of the quotient. This will give the $\tan$ ratio of each corresponding angle. (Note that $\tan90^\circ$ is undefined)
For example if we want to find the value of $\sin60^\circ$, we divide $3$ by $4$ and take the square root of $\frac34$. Then we get the result $\frac{\sqrt3}{2}$ which is indeed the value of $\sin60^\circ$.
This two tricks allow to remember the whole trigonometric table using the relation of $\sin$ and $\tan$ with other trigonometric ratios. I think the trick is well-known.
My question is that why do these tricks work? Are they kind of coincidence (which is highly unlikely)? I can prove the trigonometric ratios individually. For example, I can prove geometrically that $\sin60^\circ$ is equal to $\frac{\sqrt3}{2}$.
Best Answer
Of course it's not just coincidence. The angles $0^\circ,$ $30^\circ,$ $45^\circ,$ $60^\circ,$ and $90^\circ$ are "standard angles" precisely because they are easy to express as exact numbers of degrees and they have relatively simple expressions for sine and cosine compared to other such angles.
The angle $30^\circ$ is nice because it is one of the angles of the right triangles that you get when you cut an equilateral triangle into two right triangles. Half of one side of the equilateral triangle becomes a leg of a right triangle and another side becomes the hypotenuse, and as a result it's easy to see that $\sin(30^\circ) = \frac12.$
The square root part of the trick comes from the Pythagorean Theorem. If one acute angle of a triangle is $\theta,$ the other angle is $90^\circ - \theta,$ and it follows that
$$ (\sin(\theta))^2 + (\sin(90^\circ - \theta))^2 = 1. $$
Let $\theta = 45^\circ,$ and then $90^\circ - \theta$ is also $45^\circ$, so $(\sin(\theta))^2$ added to itself is $1.$ It follows that $(\sin(45^\circ))^2 = \frac12.$
Let $\theta = 30^\circ$; we already found that $\sin(30^\circ) = \frac12,$ so $(\sin(30^\circ))^2 = \frac14.$ But then $90^\circ - \theta$ is $60^\circ,$ so $(\sin(30^\circ))^2 + (\sin(60^\circ))^2 = 1.$ That implies that $(\sin(60^\circ))^2 = \frac34.$
Obviously $\sin(0^\circ) = 0$ and $\sin(90^\circ) = 1.$
So now we have found five simple-to-describe angles whose sines, when squared, happen to produce the numbers $0, \frac14, \frac12, \frac34, 1.$ That sequence of numbers can also be written $\frac04, \frac14, \frac24, \frac34, \frac44.$
The middle three values are in arithmetic sequence, of course, because the equation derived from the Pythagorean Theorem ensures that $(\sin(45^\circ - \delta)),$ $(\sin(45^\circ)),$ and $(\sin(45^\circ + \delta))$ will be in sequence for any $\delta,$ in this case $\delta = 15^\circ.$ But it happens that all five values are in arithmetic sequence -- due entirely to the fact that $(\sin(30^\circ))^2$ is exactly half of $(\sin(45^\circ))^2,$ which is the only "coincidental" fact in the derivation of this sequence -- and you can turn them into a sequence of consecutive integers by applying the necessary scaling, in this case multiplying by $4.$ So that's where that "trick" comes from.
For the tangents, we know the tangent is the ratio of sine and cosine, and the cosine of an angle $\theta$ is just the sine of $90^\circ - \theta,$ so the square of the tangent is a ratio of squares:
$$ (\tan(\theta))^2 = \frac{(\sin(\theta))^2}{(\sin(90^\circ - \theta))^2}. $$
Plugging in what we already know about the sines of the "standard angles," we find that $(\tan(0^\circ))^2 = 1,$ $(\tan(30^\circ))^2 = \frac13,$ $(\tan(45^\circ))^2 = 1,$ and $(\tan(60^\circ))^2 = 3.$
All of these tangent values are integers already except for $\frac13,$ so if we just multiply by $3$ we have all integers again. But the integers $0,1,3,9$ are not in any particularly nice sequence; not consecutive, certainly, not arithmetic, not even a geometric sequence (because of the $0$). The last three entries are in geometric sequence, of course, because in general $\tan(\theta) = 1/\tan(90^\circ - \theta),$ from which it easily follows that the three tangent values $\tan(45^\circ - \delta),$ $\tan(45^\circ),$ and $\tan(45^\circ + \delta)$ will always be in geometric sequence, and therefore so will their squares.
When I compare the effort of memorizing the two (different) sequences for sine and tangent (are we really supposed to find it so "simple" to remember the sequence $0,1,3,9$?), plus the effort of memorizing the two (different) scaling factors (divide by $4$ and divide by $3$ -- why?), plus the effort of remembering to take square roots at the correct step in the procedure, versus the effort of drawing a $45^\circ$ right triangle and a $60^\circ$ right triangle and applying the Pythagorean Theorem to assign lengths to all the sides of both triangles (in one case, $1,1, \sqrt2$, in the other $1,\sqrt3,2$) and then just reading off all the angle functions via SOHCAHTOA, I'm not particularly impressed by the "trick." Both methods take some work but the non-"trick" method seems much more natural and involves a lot less memorization of facts that you can never use elsewhere.
The real "trick" is in getting us to believe that these particular angles are the angles we should care about. Why should the angles be $0^\circ,$ $30^\circ,$ $45^\circ,$ $60^\circ,$ and $90^\circ$? There is a $30$-degree gap between the first two angles and the last two but only $15$-degree gaps between other angles. Why the uneven gaps? Why not adjust some of the angles to make the gaps all the same, so the angles are $0^\circ,$ $22\frac12^\circ,$ $45^\circ,$ $67\frac12^\circ,$ and $90^\circ$? Why not fill in the larger gaps in the obvious way: $0^\circ,$ $15^\circ,$ $30^\circ,$ $45^\circ,$ $60^\circ,$ $75^\circ,$ and $90^\circ$ so that all gaps are equal?
In fact these alternative "standard" angles, namely $15^\circ,$ $22\frac12^\circ,$ $67\frac12^\circ,$ and $75^\circ,$ also have sines and tangents that are relatively easy to write using integers, simple arithmetic, and the square root function, but the expressions we get are not quite as simple as the sines and cosines of the usual "standard" angles. So those alternative angles generally don't end up in your table of "standard" sine and tangent values.
In short, you can make simple rules for generating this table because any angle that would not allow simple rules has been excluded.