What are some mind-blowing facts about mathematics? by Soham Bansal

Answer by Soham Bansal:

This hexagon is a special

diagramto help you remember some Trigonometric IdentitiesSketch the diagram when you are struggling with trig identities … it may help you! Here is how:

Building It: The Quotient IdentitiesStart with:

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

To help you remember

think "tsc !"Then add:

- cot (which is
cotangent) on the opposite

side of the hexagon to tan- cosec or csc (which is
cosecant) next, and- sec (which is secant) last

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 IdentitiesThe hexagon also shows that a function

betweenany 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) = 1Some 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)

Bonus!AND we also get these:

Examples:

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

Double Bonus: The Pythagorean IdentitiesThe Unit Circle shows us that

sin^2 (x) + cos^2 (x) = 1The magic hexagon can help us remember that, too, by going clockwise around any of these three triangles:

And we have:

- sin^2(x) + cos^2(x) = 1
- 1 + cot^2(x) = csc^2(x)
- tan^2(x) + 1 = sec^2(x)
You can also travel counterclockwise around a triangle, for example:

- 1 – cos^2(x) = sin^2(x)

Hope this helps you!

What are some mind-blowing facts about mathematics?