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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:

Can you solve it? You might think it looks familiar.. almost like a quadratic equation, except the "x" has been replaced with a trig function? As it turns out, you're right on track. Read this Concept, and by the end, you'll be able to use the quadratic equation to solve for values of theta that satisfy the equation shown above.

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Solving Trigonometric Equations Using the Quadratic Formula

Guidance

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.

Example A

Solve for exact values of over the interval .

Solution:

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

Example B

Solve for values of over the interval .

Solution: Change to from the Pythagorean Identity.

and

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

Example C

Solve for values of over the interval .

Solution:

and

Guided Practice

1. Solve for over .

2. Solve for values of over the interval .

3. Solve the trigonometric equation such that over the interval .

Solutions:

1. 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).

2.

when , in the interval

when

3. 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.

Concept Problem Solution

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:

Explore More

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?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 3.5. 

Vocabulary

Quadratic Equation

Quadratic Equation

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.

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