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# Trigonometric Equations Using the Quadratic Formula

## The quadratic formula with a trigonometric function in place of the variable.

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Trigonometric Equations Using the Quadratic Formula

Solving equations is a fundamental part of mathematics. Being able to find which values of a variable fit an equation allows us to determine all sorts of interesting behavior, both in math and in the sciences. Solving trig equations for angles that satisfy the equation is one application of mathematical methods for solving equations. Suppose someone gave you the following equation:

### Quadratic Functions with Trigonometric Equations

When solving quadratic equations that do not factor, the quadratic formula is often used.

Remember that the quadratic equation is:

(where a, b, and c are constants)

In this situation, you can use the quadratic formula to find out what values of "x" satisfy the equation.

The same method can be applied when solving trigonometric equations that do not factor. The values for is the numerical coefficient of the function's squared term, is the numerical coefficient of the function term that is to the first power and is a constant. The formula will result in two answers and both will have to be evaluated within the designated interval.

#### Solving for Unknown Values

1. Solve for exact values of over the interval .

The equation will not factor. Use the quadratic formula for , .

2. Solve for values of over the interval .

Change to from the Pythagorean Identity.

and

This is the only solutions for since is not in the range of values.

3. Solve for values of over the interval .

and

### Examples

#### Example 1

Earlier, you were asked to solve an equation.

The original equation to solve was:

Using the quadratic formula, with , we get:

The solution of -3 is ignored because sine can't take that value, however:

#### Example 2

Solve for over .

You can factor this one like a quadratic.

For this problem the only solution is because sine cannot be (it is not in the range).

#### Example 3

Solve for values of over the interval .

when , in the interval

when

#### Example 4

Solve the trigonometric equation such that over the interval .

over the interval .

or or from the first expression, the second expression will not yield any answers because it is out the range of sine.

### Review

Solve each equation using the quadratic formula.

Use the quadratic formula to solve each quadratic equation over the interval .

1. What type of sine or cosine equations have no solution?

To see the Review answers, open this PDF file and look for section 3.5.

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### Vocabulary Language: English

A quadratic equation is an equation that can be written in the form $=ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real constants and $a\ne 0$.