# Finding Smallest Integer for Integer Expression with Trigonometric Functions

• ehrenfest
In summary, the conversation was about a trigonometry problem involving the sum to product identities. The goal was to find the smallest positive integer n for which a given expression would be an integer. The conversation concluded with a suggestion to use the sum to product identities to solve the problem.

#### ehrenfest

[SOLVED] Aime 2008 Ii 8

## Homework Statement

Let $a = \pi/2008$. Find the smallest positive integer n such that
$$2[\cos(a)\sin(a)+\cos(4a)\sin(2a)+\cos(9a)\sin(3a)+\cdots+\cos(n^2a)\sin(na)]$$
is an integer.

## Homework Equations

$$\cos(a+b) = \cos a \cos b- \sin a \sin b$$

$$\sin (a+b) = \sin a \cos b + \sin b \cos a$$

## The Attempt at a Solution

Can someone give me a hint please? This should only require high school math. I am not sure if the identities above are useful here or if there is a totally different method needed.

It's an AIME trig problem, which often means you have to play around with it and hope things end up canceling. You have a product which is difficult to sum, so try changing the product into a sum using the sum to product identities:

http://www.mathwords.com/s/sum_to_product_identities.htm

very nice

## 1. What is Aime 2008 Ii 8?

Aime 2008 Ii 8 is the acronym for the Asia-Pacific Conference on Applied Mechanics. It is a biennial conference that brings together researchers, scientists, and engineers from the Asia-Pacific region to discuss recent developments and advancements in the field of applied mechanics.

## 2. When and where is Aime 2008 Ii 8 being held?

Aime 2008 Ii 8 was held from July 8-10, 2008 in Taipei, Taiwan. The venue for the conference was the Taipei International Convention Center.

## 3. What is the theme of Aime 2008 Ii 8?

The theme of Aime 2008 Ii 8 was "Advances in Applied Mechanics for Sustainable Development". This theme focused on the application of mechanics in various fields such as energy, environment, transportation, and manufacturing, with the goal of promoting sustainable development.

## 4. Who can attend Aime 2008 Ii 8?

Aime 2008 Ii 8 was open to all researchers, scientists, and engineers from the Asia-Pacific region who are working in the field of applied mechanics. Students and postdoctoral researchers were also encouraged to attend the conference.

## 5. What were the key topics discussed at Aime 2008 Ii 8?

Some of the key topics discussed at Aime 2008 Ii 8 included solid mechanics, fluid mechanics, biomechanics, environmental mechanics, and computational mechanics. Additionally, there were special sessions on emerging fields such as nanomechanics and microfluidics.