Magic Hexagon for Trig Identities
|This hexagon is a special diagram
to help you remember some Trigonometric Identities
Sketch the diagram when you are struggling with trig identities ... it may help you! Here is how:
Building It: The Quotient Identities
tan(x) = sin(x) / cos(x)
To help you remember think "tsc"
|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":
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):
|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"
|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)
AND we also get these:
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:
And we have:
You can also travel counterclockwise around a triangle, for example:
- 1 - cos2(x) = sin2(x)
Hope this helps you!