Super Hexagon and Trigonometric Identities

Overview

• The super hexagon is a geometric figure that helps in memorizing trigonometric formulas and identities.
• It consists of a hexagon with diagonals and specific functions written at the vertices.

Structure of the Super Hexagon

• Hexagon Setup:

  • Draw a hexagon and join opposite vertices to create three diagonals.
  • Place the number 1 in the center.

• Vertices and Functions:

  • Begin at a vertex and write the functions in clockwise order:
    • tan, sine, cosine
  • The opposite vertex of tan is cot.
  • Remaining functions:
    • cosec opposite to sec (all c's on the right).

Basic Trigonometric Relationships

• Starting from tan:
  • tan θ = sine θ / cos θ
  • sine θ = cos θ / cot θ
  • cos θ = cot θ / cosec θ
  • cot θ = cosec θ / sec θ
  • cosec θ = sec θ / tan θ
  • sec θ = tan θ / sine θ

Anti-clockwise Direction

• Relationships still hold when moving anti-clockwise:
  • tan θ = sec θ / cosec θ
  • sine θ = tan θ / sec θ
  • cos θ = sine θ / tan θ

Reciprocal Relationships

• Functions at diagonally opposite vertices are reciprocals:
  • sine θ = 1/cosec θ
  • cos θ = 1/sec θ
  • cot θ = 1/tan θ

Complementary Angles

• Special relationship with complementary angles (sum equals 90 degrees):
  • sine θ = cos(90° - θ)
  • tan θ = cot(90° - θ)
  • sec θ = cosec(90° - θ)
  • cos θ = sin(90° - θ)

Pythagorean Identities

• Form triangles within the hexagon:

• Important identities derived:

  • sine² θ + cos² θ = 1
  • 1 + cot² θ = cosec² θ
  • tan² θ + 1 = sec² θ

• Anti-clockwise relations lead to:

  • 1 - cos² θ = sine² θ
  • cosec² θ - cot² θ = 1
  • sec² θ - 1 = tan² θ

Conclusion

• The super hexagon serves as a powerful tool for remembering numerous trigonometric formulas.
• It is vital to practice the positions, directions, and relationships of the functions.
• The video serves as a supplementary resource, emphasizing the importance of understanding the derived formulas rather than just memorizing them.