Periodicity of Trigonometric Functions: Exploring Waves of Sine Expressions

[greggory]

"Waves" of sine expression

So, I have been working with a lot of Math today(sorry if I am asking so many questions), and I found and expression. All sine functions use radians.

sin(y) + sin(y) / sin(y)

Now, assuming you start with 1, if you were to plot y on a graph with variable x increasing each time calculated, you would get something like this:

This image isn't mine, so this is just something identicle.

Can this be explained?

 

[Vorde]

Two things.

First of all, the equation can be simplified. Depending on the use of parentheses, if you mean $\ sin(y)$+$\frac{ sin(y)}{ sin(y)}$, this simplifies to $\ sin(y)+1$

If you meant $\frac{ sin(y)+ sin(y)}{ sin(y)}$, this simplifies to the number 2.

In the latter case, it is a null statement, but assuming you meant the first equation, the sine function is defined in a couple of cool ways (the easiest being the ratio of the opposite and hypotenuse of a right triangle), and it turns out when you define a function that way it repeats itself like a wave.

 

[greggory]

Thank you for the explanation. I was wondering why it did that(it was obvious, but any who).

But the expression sin(y) + sin(2*pi) / tan(y) does the same thing. Can that be explained?

 

[HallsofIvy]

Because $sin(2\pi)= 0$! And $tan(y)= sin(y)/cos(y)$ so that
$$\frac{sin(y)+ sin(2\pi)}{tan(y)}= \frac{sin(y)}{\frac{sin(y)}{cos(y)}}= sin(y)\frac{cos(y)}{sin(y)}= cos(y)$$

 

[Vorde]

And in a more general way, all of the trigonometric functions are periodic, so any combination of trig functions with also be periodic.