# How do you use the transformation formulas to go from product to sum and sum to product?

##### 1 Answer

**Main approach to solve a trig equation** : Use Trig Transformation Identities to transform it to a product of a few basic trig equations. Solving a trig equation finally results in solving a few basic trig equations.

**Transformation Trig Identities that convert Sums to Products** .

1. cos a + cos b = 2*cos (a +b)/2*cos (a - b)/2

2. cos a - cos b = -2*sin (a + b)/2*sin (a - b)/2

3. sin a + sin b = 2*sin (a + b)/2*cos (a - b)/2

4. sin a - sin b = 2*cos (a + b)/2*sin (a - b)/2

5. tan a + tan b = sin (a + b)/cos a*cos b.
6. tan a - tan b = sin (a - b)/cos a*cos b

**Example 1**. Transform f(x) = sin a + cos a to a product.

Solution. Use Identity (3) to transform f(x) = sin a + sin (Pi/2 - a) = 2

*sin (Pi/4)*sin (a + Pi/4)

**Example 2**. Transform f(x) = sin x + sin 3x + sin 2x to a product. Use Identity (3) to transform the sum (sin x + sin 3x), then put in common factor.

f(x) = (2

*sin 2a*cos a) + 2

*sin a*cos a = 2

*cos a*(2sin 3a/2*cos a/2)