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Sum to Product Formulas for Sine and Cosine

Relation of the sum or difference of two trigonometric functions to a product.

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Sum to Product Formulas for Sine and Cosine

Can you solve problems that involve the sum of sines or cosines? For example, consider the equation:

You could just compute each expression separately and add their values at the end. However, there is an easier way to do this. You can simplify the equation first, and then solve.

Sine and Cosine Sum to Product Formulas

In some problems, the product of two trigonometric functions is more conveniently found by the sum of two trigonometric functions by use of identities.

 

 

 

 

 

Here is an example: 

This can be verified by using the sum and difference formulas:

The following variations can be derived similarly:

Here are some examples of this type of transformation from a sum of terms to a product of terms.

Solve using the Sum to Product Formula

Change into a product.

Use the formula

Change into a product.

Use the formula

Change to a sum.

This is the reverse of what was done in the previous two examples. Looking at the four formulas above, take the one that has sine and cosine as a product, Therefore, and .

So, this translates to . A shortcut for this problem, would be to notice that the sum of and is and the difference is .

Examples

Example 1

Earlier, you were asked to solve 

You can easily transform this equation into a product of two trig functions using:

Substituting the known quantities:

Example 2

Express the sum as a product:

Using the sum-to-product formula:

Example 3

 Express the difference as a product:

Using the difference-to-product formula:

Example 4

Verify the identity (using sum-to-product formula):

Using the difference-to-product formulas:

Review

Change each sum or difference into a product.

Change each product into a sum or difference.

  1. Show that .
  2. Let and . Show that

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.13. 

Vocabulary

Sum to Product Formula

Sum to Product Formula

A sum to product formula relates the sum or difference of two trigonometric functions to the product of two trigonometric functions.

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