The Super Hexagon For all Trigonometric Formulae.
Many of us don't love maths. Especially trigonometry. There are lots of formulae in trigonometry. Memorizing them is a hard task for many. But can we memorize them just by keeping a hexagon in our minds? The answer is Yes! The Super Hexagon is the answer. But how...? Let's see it.
- Draw a Hexagon and draw its 3 diagonals.
- Write Tanθ as shown on the left side.
- Tanθ = sinθ / cosθ.
- Just memorize this single formula and write Sinθ and Cosθ respectively { Go clockwise }
- Cotθ is opposite to Tanθ. So write it on the opposite side of Tanθ.
- Take all C ( i.e. cos, cot, cosec) on the Right side. So take Cosecθ on the right side below Cotθ.
- Put Secθ below Tanθ on the remaining end
- Write 1 at the intersection of diagonals.
- tan(x) = sin(x) / cos(x)
- sin(x) = cos(x) / cot(x)
- cos(x) = cot(x) / csc(x)
- cot(x) = csc(x) / sec(x)
- csc(x) = sec(x) / tan(x)
- sec(x) = tan(x) / sin(x)
By going Anticlockwise, we get:
- cos(x) = sin(x) / tan(x)
- sin(x) = tan(x) / sec(x)
- tan(x) = sec(x) / csc(x)
- sec(x) = csc(x) / cot(x)
- csc(x) = cot(x) / cos(x)
- cot(x) = cos(x) / sin(x)
If a function is between any two functions then it's equal to them multiplied together.
If they are opposite, there's 1 between them
- sin(x)csc(x) = 1
- tan(x)csc(x) = sec(x)
- sin(x)sec(x) = tan(x)
We also get the "Reciprocal Identities", by going "through the 1"
- sin(x) = 1 / csc(x)
- cos(x) = 1 / sec(x)
- cot(x) = 1 / tan(x)
- csc(x) = 1 / sin(x)
- sec(x) = 1 / cos(x)
- tan(x) = 1 / cot(x)
The hexagon can help us remember that, too, by going clockwise around any of these three triangles:
- sin2(x) + cos2(x) = 1
- 1 + cot2(x) = csc2(x)
- tan2(x) + 1 = sec2(x)
You get to learn all formulae from this hexagon. That's why its called Super Hexagon
Image: MathsisFun.com
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