New insights for teaching kids Trigonometry, might help some homeschooling champion parents...

Sin, Cos, and Tan is unnecessarily taught all together, causing confusion, and difficulty memorising.

It's made worse by teaching fractions of triangle sides, "Sine is opposite over hypotenuse", and then mnemonics are taught..SOH CAH TOA.

Ridiculous.

I had to undo a lot of this crap today teaching my daughter, and the new system was easier and she got it straight away...

To begin with, forget Tan. And forget side ratios initially.

1. For any right angle triangle, the two other angles, theta and alpha, add up to 90 deg.

2. Any two right-angle (RA) triangles with the same two other angles are similar triangles. If the hypotenuse is also the same SIZE, then they are identical triangles. (Explain this by discussing zooming in and out, triangle gets bigger and smaller, but it looks like the same triangle. And so these numbers define the "shape of shapes", but not always how big or zoomed in they are).

3. Always work with triangles where the hypotenuse is 1, to begin with. To eliminate complexity of absolute sizes, start with the "shape of the shapes".

4. Sine is the main one to know, and learn to "feel" that it's the most important - it helps memory. It refers to the size of the triangle-side furtherest from an angle (don't call it a ratio yet).

Animate the angle changing, and the side's size changing with it (and triangle shape). And of course it always remains a RA triangle.

On one extreme where the angle is approaching zero the triangle becomes acute, then almost just a straight line. The hypotenuse is to be considered like a clock hand, fixed in size of 1. As the angle is approaching zero, the sine-side (opposite) is approaching 0 in length, and the other side, the cos side (adjacent), is approaching the same length as the hypotenuse, 1. Show how both sides HAVE to change size, because you're not changing the size of the hypotenuse.

On the other extreme, when the angle is opening up towards 90°, the sine side is approaching 1 in length, and the cos (adjacent) side is approaching 0. Another straight line.

When the angle is 45 deg, the animation is exactly half way between the two extremes and both angles (theta and alpha) must equal 45 degrees (otherwise it's not a right angle triangle). And the 2 changing sides are equal in length.

Demonstrating this, I think, is much more important than memorising opposite over hypotenuse, adjacent over hypotenuse, and opposite over adjacent.

Then when you have a something like, sin(37 degrees), one can appreciate this DEFINES a triangle's shape.

Then, once they get that, you can explain the vale of sine(T) is actually a COMPARISON to the hypotenuse, which has always been 1 so far, and also is always the longest side of the RA triangle (so the value of a side has always been under one). Comparison means a ratio.

0.5 means half the size compared to the hypotenuse.

THEN, show...

sine (45) = A/H = 0.7

... as an example, and demonstrates how with bigger triangles (hypotenuse greater than 1), if the shape of the triangle is the same, defined by angle, then sin(T) stays the same, and always under 1, no matter the larger hypotenuse.

Once comfortable, draw sine waves and cos waves (careful not to become the whiteboard meme guy here).

Then when comfortable with that, show a clock and the triangles formed in the 4 quadrants.

Introduce radians.

Only THEN talk about Tan, opposite over hypotenuse, or sineT/cosT.

Just my 2 π