Sum to Product Trigonometric identity does not work

[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

 

[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.

 

[micromass]

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.

 

[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.

 

[micromass]

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.

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.