Interactive Unit Circle, Built in
Trigonometry is probably the most inaptly named subject in the standard K12 math curriculum. Although we do typically introduce trigonometric ideas with triangles (specifically right triangles), trigonometry isn't really about *trigons*. It's about circles. And today, I am excited to release an interactive unit circle that can help students develop and remember certain key trigonometric facts. You can find it at https://www.desmos.com/calculator/ldd....
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Students quickly discover this when teachers start to ask about angles 90 degrees or greater, or when we start to write angle measures using this weird thing called "radians" instead. Trigonometry is in many ways better understood as circle-ometry, and there's one particular circle that stands above the rest: the unit circle.
The unit circle is a circle of radius one *unit*, centered at the origin. The values of the trigonometric ratios we learn about in the context of right triangles are in fact better understood as simple the different values of the coordinates we get as travel along the edge of the unit circle. When we say something like the sine of 30 degrees is 1/2, what we mean is that the y-coordinate of the ordered pair on the edge of the unit circle 30 degrees clockwise from the positive x-axis is 1/2. The cosine of any given angle is its x-coordinate on this same circle, and the tangent is simply the ratio of the sine value to the cosine value (so, the y-coordinate to the x-coordinate).
Viewed this way, other mnemonic devices like "All Students Take Calculus", which helps us remember which trig functions are positive in which coordinates, are really just placeholders for the quadrants themselves. Sine, for example, is positive in the first and second quadrants (the "All" and "Students" of the mnemonic device) because the y-coordinate is positive in the first and second quadrants.
Finally, although it's good to be able to generate all of this from first principles, there is a place for simple memorization, and that's the other thing teachers love to do with the unit circle. Memorize it, and you won't have to write out special right triangles every time you need a certain value. Again, if you'd like to check out the interactive unit circle I built in @Desmos for this video, you can find it at https://www.desmos.com/calculator/ldd....
#unitcircle #trigonometry #desmos
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