# Sin/Cos periodic 2*pi: I'm getting confused about how sin(x)=sinx(x + (2*pi)) why is that?

thanks alot!!!!

##### Junior Member
guys I really need help, why sinx(x)=sin(x+2*pi) ? argument aren't the same so how at specific x we get the same value of two functions?!

#### topsquark

##### Full Member
HallsofIvy had a good post. The way the function sin(x) is defined makes it periodic. If you go around the unit circle once (an angle of $$\displaystyle 2 \pi$$ radians) the you get the same value back. If you go around twice ( $$\displaystyle 4 \pi$$ ) radians you get the same value back again. Note how the graph below repeats itself every multiple of $$\displaystyle 2 \pi$$ radians.

-Dan

#### Ryan$##### Junior Member So we can have the same function with two different argument ? like sinx=sinx(x+180) and that's why called periodic? I mean periodic implicitly mean we could have different argument with the same function's value?! thanks #### Otis ##### Senior Member Hello Ryan$. It looks like you're sometimes mistyping the function's name sin as sinx.

Also, I think it's best to use function notation, when typing functions. I've made some other comments below, too.

… sin(x)=sinx(x + (2*pi)) …
We don't really need grouping symbols around 2∙pi, but I like that you used function notation on each side.

However, typing sinx(x + 2∙pi) could be interpreted as either sin(x[x + 2∙pi]) or sin(x)∙(x + 2∙pi), while what you mean is:

sin(x) = sin(x + 2∙pi)

… sinx and sin(x+180) …
This time, you used function notation only on the right-hand side. Also, it's good form to type a degree symbol next to values measured in degrees, especially in a thread containing both degree measures and radian measures. What you mean is:

sin(x) and sin(x + 180º)

… sinx(x)=sin(x+2*pi) …
Good, you used function notation throughout (and no extra grouping symbols around 2∙pi), but you mean the name sin, not sinx:

sin(x) = sin(x + 2∙pi)

… sinx=sinx(x+180) …
Hopefully, you now understand three things to be fixed here. You mean:

sin(x) = sin(x + 180º)

#### HallsofIvy

##### Elite Member
You seem to be completely misunderstanding what a "function" is! You say "isn't [it] necessary for two function like sinx and sin(x+180) to have same argument for saying that two functions are the same function?!" No, the function here is "sine" (abbreviated "sin"). A function, f, takes one number as its argument, x (the "argument" of the function), and returns another number, f(x) (the "value" of the function at that argument). $$\displaystyle f(x)$$ and $$\displaystyle f(x+ 2\pi)$$ are the same function, f, evaluated at two different arguments. Whether those values are the same or not depends on exactly what the function, f, is.

#### Otis

##### Senior Member
… we can have the same function [output] with two different [arguments]? like sin(x)=sin(x+180º) …
Yes, and we don't need periodicity for that. Many functions output the same value for different inputs. Here's an example:

f(x) = x^4 + 2x^3 - 13x^2 - 14x + 24

f(-4) = f(-2) = f(1) = f(3)

… and that's why [the sine function is] called periodic? …
No, the reason why periodic functions are called periodic involves more than just outputting the same value for different inputs. My example function f above outputs the same value for multiple inputs, but f is not periodic.

Graphically speaking, a function is periodic when its curve over each period (interval) is the same as all others. In other words, a periodic function has exactly the same behavior within each of its periods.

Here are some links to free, online trigonometry textbooks and lecture-notes.

https://openstax.org/details/books/algebra-and-trigonometry

https://open.umn.edu/opentextbooks/textbooks/algebra-and-trigonometry

#### Ryan\$

##### Junior Member
So to draw sinx is the same as sin(x+2*pi)?!

#### HallsofIvy

##### Elite Member
Yes, the graph of $$\displaystyle y= sin(x)$$ is identical to the graph of $$\displaystyle y= sin(x+ 2\pi)$$.

#### Otis

##### Senior Member
So to draw sinx is the same as sin(x+2*pi)?!
Yes. The graph of sin(x+2∙pi) is the graph of sin(x) shifted 2∙pi units to the left. The period of the sine function is 2∙pi, so the behavior of sin(x+2∙pi) within [-2∙pi,0] is exactly the same as the behavior of sin(x) in [0,2∙pi]. If we plotted both functions on the same graph from -2∙pi to 2∙pi, we would see only one curve because the two graphs are identical (i.e., they match-up perfectly).

Symbol x represents an angle. We also use symbol θ, instead of x. The animation below shows why the graphs of sine and cosine repeat their behavior every 2∙pi units (i.e., every revolution around the unit circle), using these definitions:

cos(θ) = x-coordinate of point where the terminal ray of angle θ intersects unit circle

sin(θ) = y-coordinate of point where the terminal ray of angle θ intersects unit circle

Circle cos sin

LucasVB [Public domain], via Wikimedia Commons