# 3.7: Products, Sums, Linear Combinations, and Applications

## Learning Objectives

- Use the transformation formulas to go from product to sum and sum to product.
- Derive multiple angle formulas.
- Use linear combinations to solve trigonometric equations.
- Apply trigonometric equations to real-life situations.

## Sum to Product Formulas for Sine and Cosine

In some problems, the product of two trigonometric functions is more conveniently found by the sum of two trigonometric functions by use of identities such as this one:

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

The following variations can be derived similarly:

**Example 1:** Change into a product.

**Solution:** Use the formula .

**Example 2:** Change into a product.

**Solution:** Use the formula .

**Example 3:** Change to a sum.

**Solution:** 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 .

## Product to Sum Formulas for Sine and Cosine

There are two formulas for transforming a product of sine or cosine into a sum or difference. First, let’s look at the product of the sine of two angles. To do this, start with cosine.

The following product to sum formulas can be derived using the same method:

**Example 4:** Change to a sum.

**Solution:** Use the formula . Set and .

**Example 5:** Change to a product.

**Solution:** Use the formula . Therefore, and . Solve the second equation for and plug that into the first.

. Again, the sum of and is and the difference is .

## Solving Equations with Product and Sum Formulas

**Example 6:** Solve .

**Solution:** Use the formula .

**Example 7:** Solve .

**Solution:** Use the formula .

## Triple-Angle Formulas and Beyond

By combining the sum formula and the double angle formula, formulas for triple angles and more can be found.

**Example 8:** Find the formula for

**Solution:** Use both the double angle formula and the sum formula.

**Example 9:** Find the formula for

**Solution:** Using the same method from the previous example, you can obtain this formula.

## Linear Combinations

Here, we take an equation which takes a linear combination of sine and cosine and converts it into a simpler cosine function.

, where , and .

**Example 10:** Transform into the form

**Solution:** and , so . Therefore and which makes the reference angle is or -0.927 radians. since cosine is positive and sine is negative, the angle must be a fourth quadrant angle. must therefore be or 5.36 radians.The final answer is .

**Example 11:** Solve .

**Solution:** First, transform the left-hand side into the form . and , so . From this and , which makes the angle in the first quadrant and 1.176 radians. Now, our equation looks like this: and we can solve for .

## Applications & Technology

**Example 12:** The range of a small rocket that is launched with an initial velocity at an angle with the horizontal is given by . If the rocket is launched with an initial velocity of 15 m/s, what angle is needed to reach a range of 20 m?

**Solution:** Plug in 15 m/s for and 20 m for the range to solve for the angle.

You can also use the TI-83 to solve trigonometric equations. It is sometimes easier than solving the equation algebraically. Just be careful with the directions and make sure your final answer is in the form that is called for. You calculator cannot put radians in terms of .

**Example 13:** Solve such that using a graphing calculator.

Solution: In , graph and .

Next, use **CALC** to find the intersection points of the graphs.

## Review Questions

- Express the sum as a product:
- Express the difference as a product:
- Verify the identity (using sum-to-product formula):
- Express the product as a sum:
- Transform to the form
- Solve for all solutions .
- Solve for all solutions .
- Solve for all solutions .
- In the study of electronics, the function is used to analyze frequency. Simplify this function using the sum-to-product formula.
- Derive a formula for .
- A spring is being moved up and down. Attached to the end of the spring is an object that undergoes a vertical displacement. The displacement is given by the equation . Find the first two values of (in seconds) for which .

## Review Answers

- Using the sum-to-product formula:
- Using the difference-to-product formula:
- Using the difference-to-product formulas:
- Using the product-to-sum formula:
- If , then and . By the Pythagorean Theorem, and . So, because is negative, is in Quadrant IV. Therefore, . Our final answer is .
- If , then and . By the Pythagorean Theorem, . Because and are both negative, is in Quadrant III, which means
*rad*. Our final answer is .

- Using the sum-to-product formula:
- Using the sum-to-product formula: So, either or
- Move over to the other side and use the sum-to-product formula: So
- Using the sum-to-product formula:
- Derive a formula for .
- Let .