Magic Hexagon for Trig Identities

 

This hexagon is a special diagram
to help you remember some Trigonometric Identities
  magic hexagon

Sketch the diagram when you are struggling with trig identities ... it may help you! Here is how:

Building It: The Quotient Identities

Start with:

tan(x) = sin(x) / cos(x)

tsk tsk To help you remember
think "tsc !"
magic hexagon tan(x) = sin(x) / cos(x)
   

Then add:

  • cot (which is cotangent) on the opposite
    side of the hexagon to tan
  • csc (which is cosecant) next, and
  • sec (which is secant) last
magic hexagon
To help you remember: the "co" functions are all on the right

 

OK, we have now built our hexagon, what do we get out of it?

Well, we can now follow "around the clock" (either direction) to get all the "Quotient Identities":

Clockwise
  • 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)
Counterclockwise
  • 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)

Product Identities

The hexagon also shows that a function between any two functions is equal to them multiplied together (if they are opposite each other, then the "1" is between them):

magic hexagon tan(x)cos(x) = sin(x)
Example: tan(x)cos(x) = sin(x) Example: tan(x)cot(x) = 1

Some more examples:

  • sin(x)csc(x) = 1
  • tan(x)csc(x) = sec(x)
  • sin(x)sec(x) = tan(x)

But Wait, There is More!

You can also get the "Reciprocal Identities", by going "through the 1"

magic hexagon sin(x) = 1/csc(x)   Here you can see that sin(x) = 1 / csc(x)

Here is the full set:

  • 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)

Bonus!

AND we also get these:

magic hexagon sin(x) = cos(90-x),  tan(x) = cot(90-x),  sec(x) = csc(90-x),

Examples:

  • sin(30°) = cos(60°)
  • tan(80°) = cot(10°)
  • sec(40°) = csc(50°)

Double Bonus: The Pythagorean Identities

The Unit Circle shows us that

sin2 x + cos2 x = 1

The magic hexagon can help us remember that, too, by going clockwise around any of these three triangles:

magic hexagon sin^2(x) + cos^2(x)=1

And we have:

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

You can also travel counterclockwise around a triangle, for example:

  • 1 - cos2(x) = sin2(x)

Hope this helps you!