Okay, here’s my two cents.

There are two concepts to memorize.

A) Definition: $\cos = x$, $\sin = y$, $\tan = y/x$. $x^2 + y^2 = 1$.

B)Then for specific angles there are values. These are basically geometry and there are three cases:

1) $\theta$ ~ $0$ or $90$. These are $\theta = 0, \pi/2, \pi, 3\pi/4$ (or $-\pi/4$).

These are the points $(x,y) = (\cos \theta, \sin \theta)$ = $(0, \pm 1)$ and $\tan \theta = (0, \pm 1)$ and $x/y$ = $0$,$ \pm 1/0$ (undefined).

Which is which is visually apparent by imagining the circle.

2) $\theta$ ~ 45. These are the $\theta = \text{odd}*\pi/4$ (or another way to view it $\{0, \pi, \pm \pi/2\} \pm \pi/4$; or simply $\pi/4, 3\pi/4, 5\pi/4 = -3\pi/4, 7\pi/4 = -\pi/4$).

These form 45-45-90 triangles, or an isosceles right triangle. $x = y$ and so $x^2 + y^2 = 2x^2 = 1$ so $|x| = |y| = \sqrt{2}/2$. And $|x|/|y| = 1$.

So $\sin \theta = \pm \cos \theta = \pm \sqrt{2}/2$. and $\tan \theta = \pm 1$.

Which pos/neg polarity depends upon which quandrant $(x,y) = (\cos \theta, \sin \theta)$ lies in.

3) $\theta$ ~ $30, 60$. This are $\{0,\pi, \pm \pi/2\} \pm \pi/6$. Or $0, \pi/6, \pi/3, 2\pi/3, 5\pi/6, 7\pi/6, 4\pi/3, 5\pi/3, 11\pi/6$.

These for 30-60-90 degree triangles which are equilateral triangles cut in half. As right triangles have $a^2 + b^2 = c^2$ and as $c = 1$ then $a = 1/2$ and $b = \sqrt{3} 2$.

So these are $(x,y) = (\cos \theta, \sin \theta) = (\{\pm 1/2: \pm \sqrt{3}/2\},\{\pm \sqrt{3}/2: \pm 1/2)$ and $\tan \theta = \{ \pm 1/\sqrt 3 : \pm \sqrt 3\}$

Which values depend upon whether $|x| > |y|$ or $|y| > |x|$ and which quadrant $(x,y) lie in.


And yes, that circular picture helps. As the circular picture is perfectly symmetric it’s easy to memorize even though it does, technically, have 48 values. As it’s symmetric it reduces to really only the three cases above.

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