6.1.1 Solving Trig Equations

1. 6.1.1 Solving Trig Equations Tuesday, February 11, 2014

2. Introduction • You can already solve all kinds of equations. You have been doing it since your first algebra class. • You know that some equations have no solutions, some have only one solution, and others can have any number of solutions, even an infinite number. x + 1 = x + 1 x = x + 1 5 = x + 1

3. Introduction • Today’s lesson focuses on solving trigonometric equations (which we will refer to as trig equations from now on). • The concepts and the procedures you will use are basically the same, but there are a few unique qualities of trig equations that you have to keep in mind. • Let’s see what sets this kind of solving apart from the rest.

4. 6-1. We begin with the graphs of y = sin x and y= shown below. For now, concentrate on the portion of the graph shown, 0 ≤ x ≤ 2π. • How do you know that the equation, sin x = has more than one solution? • What are the solutions for sin x= , where 0 ≤ x ≤ 2π?

5. 6.1 Continued • Now recall the unit circle. The horizontal line y = has been drawn across the circle. Explain how this can help to find solutions to the equation sin x = for 0 ≤ x ≤ 2π.

6. 6.1 Continued • Using both the graph and the unit circle, we want to solve the equation sin x = . • How?

7. 6-2. What about cos x= over the same interval, 0 ≤ x ≤ 2π? Draw a line on the unit circle so that you can find where the x-coordinate equals. What angles satisfy the equation?

8. Closure • Solving a trigonometric equation is like solving any equation: our goal is to get the variable by itself on one side of the equal sign. • We want to find out what values of the variable will make the equation “true.” • But, we need to keep in mind when solving trig equations, that there are not necessarily a finite number of solutions. • The solutions we want will depend on the problem situation. • If only a few solutions are required, we can limit the domain to a particular interval.

9. MATH NOTES - Solving Periodic Functions • When solving periodic functions (such as sin x= ), we get an infinite number of solutions (or no solution). • In the example sin x= , we get the solutions:x = … … and x = … … • This may be written more compactly as x= + 2πn, + 2πn, where n is any integer. • If we restrict the domain to [0, 2π), we get only two solutions: x= and .