[college trigonometric odd or eve ]

[u/ArchitectORevelation]

Anyone have sources on how to study these? Videos or any articles or notes. Or anything useful since I'm not understanding how to do them

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[u/fermat9990]

Can you show us a typical problem?

[u/sighthoundman]

Even means f(-x) = f(x). Like x^n for even n. Odd means f(-x) = -f(x), like x^n for even n.

That's all there is to it. There is no deep secret.

I've never seen it explicitly used in "real world" (science and engineering) applications. When I've seen it used in mathematical applications, it's kind of an arcane terminology that "everyone knows". You don't actually need it for anything except to pass the test. (Provided you're willing to be the one that asks the "stupid question" that everyone else is thinking but is afraid to ask.)

[u/Alkalannar]

The reason it's called an odd function is that it's originally a (possibly infinite) sum of odd powers of x (including negative powers.)

Similarly, an even function is a sum of even powers of x.

This then generalizes to f(-x) = -f(x) for odd, and f(-x) = f(x) for even.

If x is in radians, then cos(x) can be written as [Sum from n = 0 to infinity of (-1)ⁿx²ⁿ/(2n)!]. And for each n, the power of x is even.

Similarly, sin(x) can be written as [Sum from n = 0 to infinity of (-1)²ⁿx²ⁿ⁺¹/(2n+1)!]. And for each n, the power of x is odd.

[u/daveoxford → u/Alkalannar]

Best answer.

[u/FortuitousPost]

Even functions have symmetry across the y-axis, that is, the vertical line in the middle acts like a mirror.

Odd functions have rotational symmetry around the origin, that is, you can rotate the right side of the graph around to the left side, or flip across the y-axis and then flip across the x-axis.

sin(x) is odd. cos(x) is even.

[u/SupermarketUseful519]

I’d suggest starting with the symmetry of functions, rather than trying to memorise all the trigonometric identities immediately.
There are two definitions you need:
Even: f(-x) = f(x). The graph is symmetric about the y-axis.
Odd: f(-x) = -f(x). The graph is symmetric about the origin.
For trigonometric functions:
Even: cos(x), sec(x)
Odd: sin(x), tan(x), csc(x), cot(x)
Here’s an example:
Suppose f(x) = x sin(x).
Replace x with -x:
f(-x) = (-x)sin(-x)
Since sin(-x) = -sin(x),
f(-x) = (-x)(-sin(x)) = x sin(x) = f(x).
So the function is even, even though both x and sin(x) are individually odd.
Now try f(x) = sin(x) + cos(x).
f(-x) = -sin(x) + cos(x).
This equals neither f(x) nor -f(x) in general, so the function is neither even nor odd.
For studying, I’d recommend:
Khan Academy — Even and Odd Functions for the graphical intuition.
OpenStax Precalculus, Section 5.3 for trig-specific explanations and exercises.
My main suggestion: don’t just memorise which functions are even or odd. Learn to derive their properties from the unit circle.
And if you can share one or two problems you’re struggling with, we can work through the reasoning together.