Trigonometric Identities

Magic Hexagon for Trig Identities

• Reciprocal Identities

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

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

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)

• 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):

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

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

• Quotient Identities 

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

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)

• 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:

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