Special Angles

Algebra 2 · Axiom Academy

Where the exact values of 30°, 45°, and 60° come from — read straight off two special right triangles sitting on the unit circle. Cut a unit square along its diagonal and you get an isosceles right triangle : two equal legs meeting at a right angle. Drop that triangle onto the unit circle at and its equal legs become the point's coordinates. Pythagoras on two equal legs of length 1 the √2 that names this triangle Slice an equilateral triangle down its altitude and you get a 30-60-90 triangle , whose sides always fall in one fixed ratio. Scaled so the hypotenuse is 1, it fits the unit circle at both and . The long leg is horizontal, so cosine is the big one. The long leg is vertical, so sine is the big one. 3. One Pattern for the Whole Table Line the three sines up in order and a pattern jumps out: each is a square root over 2, with the number under the root simply counting 1, 2, 3 . Cosine is that same list reversed , and tangent is just sine over cosine. sine counts up; cosine runs the same list backward These first-quadrant values are the seeds for every angle. A reference angle — the acute angle back to the horizontal axis — reflects the same triangle into the other three quadrants; only the signs of the coordinates change. I: both +. II: sin +, cos −. III: both −, tan +. IV: cos +, sin −. You can now rebuild the exact values of every special angle from two triangles — no calculator, no rote memorization. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Algebra 2 course.