Solving Natural Sinus Problems with x=pi

[asi123]

problem with sinus :)

Homework Statement

Hey guys.
cos(n*x) when n is natural and x=pi equals to (-1)^n as I wrote in the pic.
What I'm looking for is the same thing in sinus, let's say again that n is natural, is there a x that can make sinus to behave something like cos? and by that I mean an expression that can be written in on line like (-1)^n (x can be anything).
I hope that was understood.
Thanks in advance.

Homework Equations

The Attempt at a Solution

 

[Dick]

Like sin(n*pi+pi/2)=(-1)^n?

 

[asi123]

Like sin(n*pi+pi/2)=(-1)^n?

That's the idea except that I have the expression (nx) inside the sinus and from that I can't get n*pi+pi/2 (x can not be depended in n).
lets say x=pi/2, is there a way to write the result in one line? like cos(n*pi) = (-1)^n.

 

[Borek]

sin(n*pi/2)? For any integer n? Is that what you are asking for?

If so, it is even simpler than cos. Take a look at sin plot.

 

[Dick]

Then how about sin(n*pi)=0?

 

[Borek]

sin(n*pi)... that's what I meant

Combining sinus problem with sinuses problems is a sure way of getting headache.