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360 degrees in a circle is a historical accident — probably from Babylonian astronomy. Radians are different: one radian is defined so that arc length equals radius. That definition makes the derivatives of sine and cosine work without ugly correction factors.

Every trig identity has a geometric proof hiding behind it. Once you see sine and cosine as coordinates on the unit circle, identities like sin²θ + cos²θ = 1 stop being things you memorize and start being things you can't unsee.

The mnemonic gets you through the test. But sine and cosine aren't really about triangles — they're about where a point ends up after rotating around a circle. Once you see that, the right triangle is just a special case.

The Pythagorean theorem is a special case — it assumes a 90-degree angle. The Law of Cosines handles any triangle by adding a correction term that vanishes at 90°. That's not coincidence; it's the geometry.

The Law of Sines isn't just a formula for solving triangles — it's a geometric truth: every triangle fits perfectly inside one unique circle, and the ratio of sides to sines is twice that circle's radius.

Place a point on a circle of radius 1. Its x-coordinate is the cosine of the angle; its y-coordinate is the sine. That's it. The unit circle strips trig functions down to their geometric core and makes identities like sin²θ + cos²θ = 1 visually obvious.

Sine and cosine measure horizontal and vertical position; tangent measures their ratio. That ratio is the slope of a line at a given angle — which is why tangent explodes to infinity at 90°, where the line becomes vertical. Understanding tangent geometrically makes its behavior intuitive.