# Trigonometric Polynomials

**Trigonometric Polynomials:
**

A Trigonometric Polynomial is a polynomial expression involving** cos x** and **sin x:**

cos^{5}x 6 cos^{3}x sin^{2}x 3 sin^{4}x 2 cos^{2}x 5

Because of the identity cos^{2}x sin^{2}x = 1, most trigonometric polynomials can be written in several dierent ways. For example, the above polynomial can be rewritten as:

5 cos^{3}x sin^{2}x 3 sin^{4}x cos^{3}x - 2 sin^{2}x 7

**Fourier Sums:**

A Fourier Sum is a Fourier Series with nitely many terms:

5 3 sin^{ 2}x 4 cos^{ 5}x - 3 sin ^{5}x 2 cos^{ 8}x:

Every Fourier sum is actually a trigonometric polynomial, and any trigonometric polynomial can be expressed as a Fourier sum. Converting a Fourier sum to a trigonometric polynomial is fairly straight forward: simply substitute the appropriate multiple-angle identity for each cos nx and sin nx (see table 1). It is less obvious that every trigonometric polynomial can be expressed as a Fouriersum. This depends on the three **product-to-sum formulas**:

These identities allow us to transform any product of trigonometric functions into a sum. By applying them repeatedly, we can remove all of the multiplications from a trigonometric polynomial, resulting in a Fourier sum. Alternatively, one can use these identities to derive** power-reduction formulas **for cos^{j}x sin^{k}x, the first few of which are listed below:

These formulas tell us how to convert each term of a trigonometric polynomials directly into Fourier Sum.