SOH-CAH-TOA is training wheels — the unit circle is the bike

Every student arrives knowing SOH-CAH-TOA, and for a while it's enough. Opposite over hypotenuse, adjacent over hypotenuse — find the missing side, find the missing angle, done. Then the syllabus asks for $\sin 150°$, or the sine rule produces an obtuse angle, and the mnemonic just... stops. There is no triangle with a 150° angle inside a right-angled triangle. The training wheels come off and the student falls.

The fix isn't a better mnemonic. It's a better picture.

Sine and cosine are coordinates, not ratios

Draw a circle of radius one, centred on the origin. Take a point on it and let $\theta$ be the angle from the positive $x$-axis. Then, by definition:

That's the whole idea. Cosine is how far across the point is; sine is how far up. For an angle inside a right-angled triangle this agrees perfectly with SOH-CAH-TOA — but it keeps working when the angle doesn't fit in a triangle at all.

Suddenly the hard things are obvious

Once sine is just "height on the circle," a pile of facts students used to memorise turn into things they can simply see:

• Why $\sin 150° = \sin 30°$. Both points sit at the same height on the circle, one on the right, one on the left. Same $y$, same sine.
• Why cosine is negative on the left. Past 90°, the point is to the left of the origin, so its $x$-coordinate is negative. Of course cosine is.
• Why $\sin$ and $\cos$ never exceed one. The point lives on a circle of radius one. You can't be more than one unit up.

A student who pictures the circle never has to ask "is it positive or negative in this quadrant?" They just look at where the point is.

I still teach SOH-CAH-TOA — it's the right first step, and it's genuinely useful for plain triangles. But I introduce the circle early and keep coming back to it, because the day the angles leave the triangle, it's the only thing left standing. It's also the picture that makes radians finally make sense when they turn up a year later.