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Solving Trigonometric Equations

on the TI-89

The TI-89 has a very powerful built-in "solve(" function, which you will find very handy - if you know how to use it intelligently.

Consider Example 6 on page A30 in the Stewart text: "Find all values of x in the interval [0, 2PI] such that sin x = sin 2x."

The result of entering this function into the TI-89's solve function as "solve(sin(x)=sin(2x),x)" is shown at right. (You can get the solve function by pressing F2 and selecting "1: solve(" or by typing the letters).

The calculator has delivered several solutions ( the right-pointing arrow on the solution line indicates that there are more solutions on the line), but they are approximations. This may or may not be good enough.

solve(sin(x) = sin(2x),x)

In order to help the TI-89 find exact solutions, we can follow the lead of Example 6 and realize that sin 2x = 2 sin x cos x, which reduces all of the functions to the same argument.

Entering "solve(sin(x)=2sin(x)cos(x),x)" produces the result shown at right.

Well, the good news is that the TI-89 has produced exact solutions for this version of the equation. The bad news is that you probably have no idea what they are. Here's the deal: In the expression (6@n1 + 1)PI/3, @n1 represents an integer, so, solutions indicated by this expression are:

trig equation solutions

(Notice that only one of these solutions is in the domain of the problem.) In this way, you can find all of the solutions that fall in the interval .


Is there an easier way? Yes, sometimes. The screen shot at right shows what happens when you specify the domain of solution to the solve command. Here, adding | x >= 0 and x <= 2PI gives the 5 solutions in the interval in easily-readable form. Hooray!

Note: You get the "greater than or equal" sign by pressing Diamonddecimal point (instead of 2nddecimal point), and you can get "and" from CATALOG or by typing the letters.

specifying the domain

last update August 17, 2005 by JL Stanbrough