Summary
Highlights
The Super Hexagon is a simple hexagon with opposite vertices joined by three diagonals, and the number one placed in the center. The six trigonometric functions are then strategically placed at each of the six vertices. The placement starts with 'tan', 'sine', 'cos' in clockwise direction, 'cot' opposite 'tan', and 'cosec' on the right side with all 'c' functions, leaving 'sec' for the last vertex.
By moving clockwise from any function, the first function equals the second function divided by the third function (e.g., tanθ = sinθ/cosθ). This pattern generates six formulae. The same principle applies when moving anti-clockwise (e.g., tanθ = secθ/cosecθ), generating another six formulae.
Multiplying the functions at opposite vertices always results in 1. For example, sinθ × cosecθ = 1, cosθ × secθ = 1, and tanθ × cotθ = 1. This reveals three important reciprocal relationships.
For any three continuous functions in the hexagon, the product of the first and third functions equals the middle function (e.g., tanθ × cosθ = sinθ). This pattern can be applied throughout the hexagon.
Functions at diagonally opposite vertices are reciprocals of each other. Arrows drawn between these pairs indicate that one function is 1 divided by the other (e.g., sinθ = 1/cosecθ and cosecθ = 1/sinθ).
Horizontal arrows in the hexagon represent complementary angle relationships. For instance, sinθ = cos(90° - θ), tanθ = cot(90° - θ), and secθ = cosec(90° - θ). Reversing the arrows gives similar identities.
Focusing on specific triangles formed within the hexagon and moving clockwise (starting from the top-left function), allows for the derivation of Pythagorean identities (e.g., sin²θ + cos²θ = 1, 1 + cot²θ = cosec²θ, tan²θ + 1 = sec²θ). Moving anti-clockwise introduces a minus sign (e.g., 1 - cos²θ = sin²θ).
The Super Hexagon is a powerful mnemonic for remembering numerous trigonometric formulae and identities. However, it requires careful practice to correctly recall the positions of functions and the directions for applying the rules. The video emphasizes that this tool is an add-on and not a substitute for understanding the underlying concepts of trigonometry.