Is the Sine Function Linear in Small Domains?

[Cluelessness]

Sine function linear??

I have a problem concerning trigonometry and calculus but I just need to know my question's answer to solve it.
I would like to know: can a sine function be construed as a linear function in a very small domain i.e increments of 0.0001??
Thank you so much in advance and I appreciate all your help :D

 

[Dustinsfl]

I have a problem concerning trigonometry and calculus but I just need to know my question's answer to solve it.
I would like to know: can a sine function be construed as a linear function in a very small domain i.e increments of 0.0001??
Thank you so much in advance and I appreciate all your help :D

If sine was linear, then $\sin(x+y) = \sin x + \sin y$, but
$$
\sin (x+y) = \sin x\cos y + \sin y\cos x
$$
So that would only occur if $\cos y = \cos x$.

Therefore, $y = 2\pi k$ and $x = 2\pi n$ where $k,n\in\mathbb{Z}$.

The case when $\sin (x+y) = 0$ would be linear.

 

[SammyS]

I have a problem concerning trigonometry and calculus but I just need to know my question's answer to solve it.
I would like to know: can a sine function be construed as a linear function in a very small domain i.e increments of 0.0001??
Thank you so much in advance and I appreciate all your help :D

Depending upon the details of whatever situation is being addressed, it would generally be acceptable to treat the sine function as being linear over a span of 0.0001 of a single period of the sine function.

 

[nucl34rgg]

sin(x) is approximately equal to x for "small x," but I don't think this is what you were asking.

I think your intuition is correct in that most continuous functions can be well approximated and built by small linear increments. This is basically the idea behind Euler's method. http://en.wikipedia.org/wiki/Euler_method#Informal_geometrical_description