2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measure angle traversed counterclockwise around the unit circle.

We can't expect our best friends to like and dislike all the same people we do. Everyone has different opinions, and we're all entitled to them. But if your best friend hangs out with your sworn enemy, things are bound to get a little bumpy. Just remember: you can never have too many friends.

So when students find out that their best friend, the unit circle, has been fraternizing with not one, not two—but six trigonometric functions, their initial reaction might be one of utter shock, horror, and despair.

But…maybe this unforgivable betrayal is really an opportunity in disguise. Sure, trigonometric functions haven't been the easiest to deal with, but there's no need for a Sharks-versus-Jets rivalry, is there? After all, there must be a reason the unit circle hangs around with trig functions.

Students should use the unit circle to define all six trigonometric functions in terms of coordinates x and y and radius r. A reference triangle with x and y coordinates and radius r, and angle θ is a good place to start.

Hopefully, students are already proficient using the Pythagorean rule and basic trigonometry. If they've forgotten just about everything, it might be good to remind them of SOHCAHTOA. At least that way, they can look trig functions in the eye again.

Once students know the three main functions (sine, cosine, and tangent), teachers can help students remember reciprocal functions by always pairing functions with their reciprocals (sine with cosecant, cosine with secant, and tangent with cotangent). At first, anyway.

Start in the neutral territory of quadrant I where everything is positive, and slowly start venturing elsewhere. The mnemonic "All Students Take Calculus" (or whatever other mnemonic you think of for ASTC) often helps students remember which of the six trigonometric functions is positive in each of the four quadrants.

Drawing simple sketches in the coordinate plane with angles in all four quadrants will help students learn how to find reference angles. Starting in degrees may help make the math theory less intimidating for students. Once the students are familiar with reference angles, it's easy to transition to radians.

It won't happen overnight, but eventually students and trigonometric functions will start spending more time together. Sometimes, they won't even need the unit circle to mediate between them. And whether they know it or not, it'll be the start of a beautiful friendship.