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# Trigonometric Equations Using Factoring

## Factoring and the Quadratic Formula.

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Solving Trigonometric Equations Using Quadratic Techniques

As Agent Trigonometry, you are given this clue: . If falls in the interval , what is/are its possible value(s)?

### Guidance

Another way to solve a trig equation is to use factoring or the quadratic formula. Let’s look at a couple of examples.

#### Example A

Solve .

Solution: This sine equation looks a lot like the quadratic which factors to be and the solutions are and 1. We can factor the trig equation in the exact same manner. Instead of just , we will have in the factors.

There is no solution for and when .

#### Example B

Solve in the interval .

Solution: To solve this equation, use the Pythagorean Identity . Solve for either cosine and substitute into the equation.

Solving each factor for , we get and and .

#### Example C

Solve in the interval .

Solution: This equation is not factorable so you have to use the Quadratic Formula.

and

The first answer is within the range, but the second is not. To adjust -0.954 to be within the range, we need to find the answer in the second quadrant, .

Concept Problem Revisit We can solve this problem by factoring.

.

Over the interval , when and when .

### Guided Practice

Solve the following trig equations using any method in the interval .

1.

2.

3.

Answers

1. Put everything onto one side of the equation and factor out a cosine.

2. Recall that from the Negative Angle Identities.

3. This quadratic is not factorable, so use the quadratic formula.

The other solutions in the range are and .

### Explore More

Solve the following trig equations using any method. Find all solutions in the interval . Round any decimal answers to 4 decimal places.

Using your graphing calculator, graph the following equations and determine the points of intersection in the interval .

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 14.11.

### Explore More

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