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

[Ryan$]

Hi guys, I'm studying sin/cos functions and I'm getting confused about how sin(x)=sinx(x + (2*pi)) why is that? I know that function sin is periodic at ever 2*pi, but what does that mean?! however the argument of sin is not the same(x != x+(2*pi)) we are getting the same function/answer? may please anyone illustrate for me what does "periodic" mean in math?
for instance I can assume that a specific function is periodic at every 3*pi so I can say f(x)=f(x+3*pi) ? if so .. why?! how two functions having different argument are same function?!!! the point behind the term "periodic" isn't understandable at all for me.. any help please?! thanks.

 

[HallsofIvy]

You say "I know that function sin is periodic at ever 2*pi". That is exactly what "periodic" means: sin(x+ 2pi)= sin(x). The function value repeats "periodically"- at regular intervals.

You also say "however the argument of sin is not the same(x != x+(2*pi)) we are getting the same function/answer?". The fact that $\displaystyle x\ne y$ does NOT necessarily mean that f(x) cannot be equal to f(y). Look at the very simple, constant function "f(x)= 3 for all x". That is a perfectly valid, though very simple, function. Look at $\displaystyle y= x^2$. f(-3)= 9 and f(3)= 9. The fact that the two "x" values are different doesn't mean the function values must be different.

Finally, exactly what definition of "sine" and "cosine" were you given? The definitions in terms of right triangles won't work since an angle, in a right triangle must be positive and can't be larger than pi/2 radians so "x+ 2pi" wouldn't even make sense. Most common is the "circle definition" (and some texts use the phrase "circular functions" rather than "trigonometric functions"): Draw the unit circle on a coordinate system-center at the origin, radius 1. Starting from the point (1, 0) measure a distance "t" around the circumference of the circle. The (x, y) coordinates of the end point give sine and cosine- cos(t) is the x coordinate, sin(t) is the y coordinate.

For example, since the circle has radius 1, it has circumference 2pi(1)= 2pi. pi/2 is 1/4 of that. If we start at (1, 0) and measure distance pi/2 around the circumference we go 1/4 of the way, ending at (0, 1). So cos(pi/2)= 0, sin(pi/2)= 1. If instead we measure distance pi, we go half way around the circle, from (1, 0) to (-1, 0). cos(pi)= -1, sin(pi)= 0. A little harder is sin(pi/4) and cos(pi/4). Since that is half of pi/2, we wind up half way between (1, 0) and (0, 1). By symmetry we are on the line y= x. The equation of the unit circle is x^2+ y^2= 1. Since y= x, we have x^2+ x^2= 2x^2= 1 so x^2= 1/2 and x= sqrt(1/2)= sqrt(2)/2 (we are still in the first quadrant so x and y are both positive). That is, cos(pi/4)= sin(pi/4)= sqrt(2)/2.

But, again, the entire circle has circumference 2pi. If I measure a distance around the circle x+ 2pi, I go from (1, 0) to (cos(x), sin(x)) and then on another 2pi. I have measured a distance x+ 2pi so "by definition" I must end at the point (cos(x+ 2pi), sin(x+ 2pi). But that last 2pi takes me exactly once around the circle so I come right back to (cos(x), sin(x)). So cos(x+ 2pi)= cos(x), sin(x+ 2pi)=sin(x).

 

[Ryan$]

thanks alot!!!!

 

[Ryan$]

but once again, isn't necessary for two function like sinx and sin(x+180) to have same argument for saying that two functons are the same function?! really weird

 

[Ryan$]

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]

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$]

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]

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)

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[HallsofIvy]

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]

… 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.

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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

http://www.freebookcentre.net/Mathematics/Trigonometry-Books-Download.html

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[Ryan$]

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

 

[HallsofIvy]

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

 

[Otis]

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

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