The Main Idea

The product-to-sum formulas allow us to rewrite products of sine and cosine as sums or differences. This is especially helpful in simplifying trig expressions, evaluating integrals, or solving equations. By converting products into sums, we reduce complexity and often expose familiar angles or identities.

The formulas are:

• [latex]\sin A \cos B = \dfrac{1}{2}\left[\sin(A+B) + \sin(A-B)\right][/latex]

• [latex]\cos A \cos B = \dfrac{1}{2}\left[\cos(A+B) + \cos(A-B)\right][/latex]

• [latex]\sin A \sin B = \dfrac{1}{2}\left[\cos(A-B) - \cos(A+B)\right][/latex]

Quick Tips: Using Product-to-Sum Formulas

1. Pick the Right Formula

   • Look at whether you have sine × sine, cosine × cosine, or sine × cosine.

2. Apply the Identity

   • Replace the product with the sum or difference of sines or cosines.

   • [latex]= \dfrac{1}{2}\left[\cos(90^\circ)+\cos(50^\circ)\right][/latex].

   • [latex]= \dfrac{1}{2}\left[0+\cos(50^\circ)\right]=\dfrac{1}{2}\cos(50^\circ)[/latex].

The Main Idea

The sum-to-product formulas let us rewrite sums or differences of sines or cosines as products. This is the reverse of the product-to-sum process. These formulas are especially helpful for simplifying trig expressions and solving equations because they turn a sum of two terms into a single product, often making factoring and solving easier.

The formulas are:

• [latex]\sin A + \sin B = 2\sin!\left(\dfrac{A+B}{2}\right)\cos!\left(\dfrac{A-B}{2}\right)[/latex]

• [latex]\sin A - \sin B = 2\cos!\left(\dfrac{A+B}{2}\right)\sin!\left(\dfrac{A-B}{2}\right)[/latex]

• [latex]\cos A + \cos B = 2\cos!\left(\dfrac{A+B}{2}\right)\cos!\left(\dfrac{A-B}{2}\right)[/latex]

• [latex]\cos A - \cos B = -2\sin!\left(\dfrac{A+B}{2}\right)\sin!\left(\dfrac{A-B}{2}\right)[/latex]

Quick Tips: Using Sum-to-Product Formulas

1. Identify the Pattern

   • Look for two sine terms or two cosine terms added or subtracted.

   • Match to the correct sum-to-product formula.

2. Why It’s Useful

   • Converts tricky sums into manageable products.

   • Especially useful in solving trig equations and in calculus integrals.