What is the Sum of Discrete Sinusoids?

[Superman1271]

Homework Statement

Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero.

Homework Equations

sin[8*\pi*n/5]+sin[12*\pi*n/5]

n is an integer.

The Attempt at a Solution

n----sin[8*\pi*n/5]----sin[12*\pi*n/5]

0 ---- 0------------------------------ 0

1 ---- -0.9511------------------------ 0.9511

2 ---- -0.5878------------------------ 0.5878

3 ---- 0.5878------------------------ -0.5878

4 ---- 0.9511------------------------ -0.9511

5 ---- 0------------------------------- 0

I am looking for an analytic solution thank you.

 

[Dick]

Homework Statement

Hi Everyone, I am trying to show why the given sum is zero. I am pretty sure it is zero.

Homework Equations

sin[8*\pi*n/5]+sin[12*\pi*n/5]

n is an integer.

The Attempt at a Solution

n----sin[8*\pi*n/5]----sin[12*\pi*n/5]

0 ---- 0------------------------------ 0

1 ---- -0.9511------------------------ 0.9511

2 ---- -0.5878------------------------ 0.5878

3 ---- 0.5878------------------------ -0.5878

4 ---- 0.9511------------------------ -0.9511

5 ---- 0------------------------------- 0

I am looking for an analytic solution thank you.

It's because the point between those two numbers is 10*pi*n/5=2*pi*n. Enough of a hint?

 

[HallsofIvy]

The point exactly half way between those two numbers is 2pi n.

 

[Superman1271]

So on the unit circle the halfway point is always along the positive x-axis. And if that is halfway, the y components will always be the negative of each other?

Also how did you realize it was exactly half way?

 

[Dick]

So on the unit circle the halfway point is always along the positive x-axis. And if that is halfway, the y components will always be the negative of each other?

Also how did you realize it was exactly half way?

Thinking about where the points for various n lie on the unit circle is a good way.

 

[Superman1271]

Ok, Thanks =]