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Kitty Trigonometry
Here's a neat trick

Trig Learning Aid
A number of trig identities can be found in the image
Kitty demonstrated the study aid to her 3rd period precalculus class



Kitty, (her nickname), is an exchange student visiting Southwest from China.  She demonstrated to her third period class a very slick memory device for helping a person recall a number of usefull trigonometric identities.  I have been studying and teaching math for many years and had not seen this wonderful learning aid.


  • The diagram is drawn with a side (rather than a vertex) at the top.
  • Each function's reciprocal function is on the opposite side of the wheel. 
  • Each of the functions with the prefix "co" appear together on the right-hand side.  

Reciprocal Identities:

The product of any trig function and the function directly across from it is 1.  This is because these pairings are reciprocals of each other.

  • sin(x) · csc(x) = 1
  • cos(x) · sec(x) = 1
  • tan(x) · cot(x) = 1

Pythagorean Identities:

The sum of the squares of the functions on the top of the shaded triangles will equal the square of the function at the bottom of the shaded triangle.

  • sin²( x) + cos²( x) = 1
  • tan²( x) +   1      = sec²( x)
  •     1    + cot²(x)  = csc²( x)

Product of Trig Functions Identities:

A trig function is equal to the product of its two neighbors.

  • sin(x) = tan(x) · cos(x)
  • cos(x) = sin(x) · cot(x)
  • cot(x) = cos(x) · csc(x)
  • csc(x) = cot(x) · sec(x)
  • sec(x) = csc(x) · tan(x)
  • tan(x) = sec(x) · sin(x)

Perhaps there are more identities hidden away.