# Sum to Product Trigonometric identity does not work

1. Apr 27, 2013

### CraigH

"Sum to Product" Trigonometric identity does not work

Hi,

The identity

$sin(u) + sin(v) = 2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2})$
http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Product-to-sum_and_sum-to-product_identities

Does not always work. I put the equation :

$(sin(u) + sin(v)) - (2 * sin (\frac{u+v}{2}) * cos(\frac{u-v}{2}))$

With u equal to -4.1 and v equal to 99 into wolfram alpha and it gave me the answer -1.11022x10^-16
http://www.wolframalpha.com/input/?i=x%3D%28sin%28-4.1%29%2Bsin%2899%29%29-%28%282*sin%28%28-4.1%2B99%29%2F2%29*cos%28%28-4.1-99%29%2F2%29%29%29

If the identity is true, shouldn't the answer always be 0?

What's going on here?

Thanks

Last edited: Apr 27, 2013
2. Apr 27, 2013

### Infrared

It seems like a rounding error to me. You can verify that the identity is in fact always true by using the half angle and angle addition formulae.

3. Apr 27, 2013

### micromass

Staff Emeritus
The problem is not that the formula doesn't work, but with the fact that your calculator is incapable of precisely calculating the sine or cosine of an angle.

4. Apr 27, 2013

### CraigH

Ah okay, thank you for answering.
One thing though... If wolfram alpha knows that it can only calculate the sine or cosine of an angle to a certain precision, shouldn't it give the final answer to that precision, or less, so that it avoids giving misleading answers like the one it gave me.

5. Apr 27, 2013

### micromass

Staff Emeritus
It would be better if they did that. But I've never seen a calculator doing it. They rather count on the users to know about the fallibility of the program.