Right Triangles and Trigonometry
Sine, cosine, and tangent all come from one idea: a point moving around a circle of radius 1. Everything else — radians, special triangles, exact values — builds from there.
This matches the 45-45-90 triangle (sides 1 : 1 : √2): both legs are equal, so sin = cos = 1/√2 = √2/2 — see the Special Triangles tab.
Picture a point moving around a circle of radius 1, centered at the origin. At angle θ (measured counterclockwise from the positive x-axis), the point’s coordinates are exactly (cos θ, sin θ) — that’s the definition, not a coincidence. The horizontal distance from the center is cosine; the vertical distance is sine.
Tangent is just their ratio: tan θ = sin θ / cos θ. It’s undefined wherever cos θ = 0 (at 90° and 270°), since you’d be dividing by zero.
Radians measure angle by arc length instead of degrees: one full circle is 2π radians instead of 360°. To convert, multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees.