High School: Functions
Trigonometric Functions HSF-TF.A.3
3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π⁄3, π⁄4 and π⁄6, and use the unit circle to express the values of sine, cosine, and tangent for π – x, π + x, and 2π – x in terms of their values for x, where x is any real number.
Students should know of the incredible magic that 30°-60°-90° and 45°-45°-90° triangles hold within them. While these triangles can't perform a simple levitation charm let alone destroy horcruxes to defeat He-Who-Must-Not-Be-Named, they're still useful because the ratios of their sides are consistent.
In math? Very helpful. In potions with Professor Snape? Not so much.
These triangles coupled with SOHCAHTOA and ASTC are like a mental cheat-sheet for finding the trig functions of π⁄3, π⁄4 and π⁄6 (or 60°, 45°, and 30°, if you're still in degree-mode). Except it's not cheating. It's magic.
It will probably help students to find the trig functions of these triangles in all four quadrants. Drawing these triangles in all four quadrants means they better get nice and close with reference angles. Yes, they need to be able to find the reference angles for π⁄3, π⁄4 and π⁄6 in quadrants II, III, and IV.
These special triangles are just more specific cases of a triangle in the unit circle. If students are confused (and chances are they will be), we suggest backtracking to the last place the students understood what was going on. This might mean repeating some concepts about radians, unit circles, or trig functions. Be patient.
Students should also know that their angle must fall within the range 0 ≤ α ≤ 2π. They need to add or subtract 2π until their angle is in the correct range. Then, they can find the right reference angle x using the formulas α = π – x, α = π + x, or α = 2π – x depending on the quadrant.
Students should be able to see that the absolute value of the trigonometric functions for π⁄3, π⁄4 and π⁄6 are determined directly from their reference triangles and that the signs of the values are determined from ASTC. These triangles will be their patronuses against the dementors of trigonometry, so help them feel the magic.