Product-to-Sum and Sum-to-Product Formulas

Section 3.4 Product-to-Sum and Sum-to-Product Formulas

In this section, we will learn how to convert sums of trigonometric functions to products of trigonometric functions, and vice versa. These techniques provide us with tools to simplify expressions and solve equations.

🔗

Subsection 3.4.1 Product to Sum Formulas

🔗

Definition 3.4.1. Product to Sum Formulas.

\begin{aligned} \sin α \cos β & = \frac{1}{2} [\sin (α + β) + \sin (α - β)] \\ \cos α \sin β & = \frac{1}{2} [\sin (α + β) - \sin (α - β)] \\ \cos α \cos β & = \frac{1}{2} [\cos (α + β) + \cos (α - β)] \\ \sin α \sin β & = \frac{1}{2} [\cos (α - β) - \cos (α + β)] \end{aligned}

🔗

Proof.

We add the addition and subtraction formulas for cosine:

\begin{aligned} \cos (α + β) & = \cos α \cos β - \sin α \sin β \\ + \cos (α - β) & = \cos α \cos β + \sin α \sin β \\ \cos (α + β) + \cos (α - β) & = 2 \cos α \cos β . \end{aligned}

Dividing both sides by 2, we get

\frac{1}{2} [\cos (α + β) + \cos (α - β)] = \cos α \cos β .

🔗

Next, we add the addition and subtraction formulas for sine:

\begin{aligned} \sin (α + β) & = \sin α \cos β + \cos α \sin β \\ + \sin (α - β) & = \sin α \cos β - \cos α \sin β \\ \sin (α + β) + \sin (α - β) & = 2 \sin α \cos β . \end{aligned}

🔗

Dividing both sides by 2, we get

\frac{1}{2} [\sin (α + β) + \sin (α - β)] = \sin α \cos β .

🔗

Next, we subtract the addition and subtraction formulas for sine:

\begin{aligned} \sin (α + β) & = \sin α \cos β + \cos α \sin β \\ - \sin (α - β) & = \sin α \cos β - \cos α \sin β \\ \sin (α + β) - \sin (α - β) & = 2 \cos α \sin β . \end{aligned}

🔗

Dividing both sides by 2, we get

\frac{1}{2} [\sin (α + β) - \sin (α - β)] = \cos α \sin β .

🔗

Finally, we subtract the addition and subtraction formulas for cosine:

\begin{aligned} \cos (α - β) & = \cos α \cos β + \sin α \sin β \\ - \cos (α + β) & = \cos α \cos β - \sin α \sin β \\ \cos (α - β) - \cos (α + β) & = 2 \sin α \sin β . \end{aligned}

🔗

Dividing both sides by 2, we get

\frac{1}{2} [\cos (α - β) - \cos (α + β)] = \sin α \sin β .

🔗

Example 3.4.2.

Express the product of

\cos (2 x) \cos (5 x)

as a sum or difference of sine and cosine with no products.

Solution.

Using the formula we get

\begin{aligned} \cos (2 x) \cos (5 x) & = \frac{1}{2} [\cos (2 x + 5 x) + \cos (2 x - 5 x)] \\ = \frac{1}{2} [\cos (7 x) + \cos (- 3 x)] . \end{aligned}

🔗

This satisfies the requirement of expressing the product of

\cos (2 x) \cos (5 x)

as a sum or difference of sine and cosine with no products. However, we can simplify it further.

🔗

Since cosine is an even function,

\cos (- 3 x) = \cos (3 x) .

Thus, we can simplify the expression to:

\cos (2 x) \cos (5 x) = \frac{1}{2} [\cos (7 x) + \cos (3 x)] .

🔗

Example 3.4.3.

Express the product of

\sin (6 θ) \cos (4 θ)

as a sum or difference of sine and cosine with no products.

Solution.

Using the formula we get

\begin{aligned} \sin (6 θ) \cos (4 θ) & = \frac{1}{2} [\sin (6 θ + 4 θ) + \sin (6 θ - 4 θ)] \\ = \frac{1}{2} [\sin (10 θ) + \sin (2 θ)] . \end{aligned}

🔗

Remark 3.4.4. Negative Angles.

In the previous example, both forms

\cos (2 x) \cos (5 x) = \frac{1}{2} [\cos (7 x) + \cos (- 3 x)]

and

\cos (2 x) \cos (5 x) = \frac{1}{2} [\cos (7 x) + \cos (3 x)]

are valid representations of the answer. However, it is more common to write the form where the angles are positive because it simplifies the expression and aligns with standard conventions for representing trigonometric identities. Positive angles are often preferred for clarity and consistency in mathematical notation. Negative angles can be transformed into positive angles using the even-odd properties of trigonometric functions (Definition 1.5.22).

🔗