Classroom Tips and Techniques: Stepwise Solution of a Trig Equation 

Robert J. Lopez 

Emeritus Professor of Mathematics and Maple Fellow 

Maplesoft 

Introduction 

 

Not long ago, I was asked how the "Equation Manipulator" could be used to provide a stepwise solution of the equation , where is given by 

 

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The "Equation Manipulator" is the tool that pops up when "Manipulate Equation" is selected in the Context Menu for an equation.  For example, making this selection for the equation 

 

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leads to Figure 1, where we have implemented the complete stepwise solution.  The drop-down box labeled "Add" provides the negative of the terms Maple sees in the equation.  The results of adding and and multiplying by are shown in Figure 1.  Note the "Undo" button on the left, and the "Return Steps" button on the bottom. 

 

Figure 1   The equation solved stepwise with the "Equation Manipulator"

 

Figure 2 shows what is returned when the "Return Steps" button is pressed. 

 

Figure 2   Stepwise solution of returned by the "Equation Manipulator"

 

Not all the steps in the manual solution of the given trig equation can be performed in the "Equation Manipulator."  Hence, we will show how a stepwise solution of this solution can be obtained in a syntax-free manner. 

 

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Preliminary Investigations 

 

Figure 3 shows a graph of the left-hand side of the equation, suggesting that there are two solutions in the interval and an infinite number of solutions on the real line. 

 

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Figure 3   Graph of for the trig equation .

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Maple provides solutions in several ways.  For example, the solve command immediately yields 

 

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(2.1)

 

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Because contains floating-point numbers, the solve command generates numeric solutions, and these include the two real solutions Figure 3 suggests lie in the interval  The Context Menu applied to the equation leads to the same results if the Solve option "Obtain Solutions for " is selected. 

 

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Stepwise Solution of the Trig Equation 

 

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Enter the equation	(3.1)
Add to both sides	(3.2)
Use the evaluation template from the Expression Palette to   make the substitution	(3.3)
Using the Context Menu, bring the resulting equation into the "Equation Manipulator".    Square both sides.    Add to both sides.    Expand the parentheses.    Press "Return Steps" button.    Via the Context Menu, select the left-hand side of the equation.    Via the Context Menu, factor this left-hand side of the equation.
Copy the first factor (linear in ), set it equal to zero, and use the Context Menu to solve for
Copy the second factor (linear in ), set it equal to zero, and use the Context Menu to solve for
Copy the factor quadratic in and use the Plot Builder to sketch Figure 4.	(3.4)

 

The graph in Figure 4 has been obtained via the plot command, but could easily be obtained with the Plot Builder accessed through the Context Menu.  The figure shows that the factor quadratic in contributes no real solutions to the solution set of the given trig equation. 

 

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Figure 4   Graph of factor quadratic in

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In Pursuit of an Exact Solution 

 

While writing the previous sections, it seemed like a good idea to convert the equation to have rational coefficients, and to articulate the steps of the solution in exact arithmetic.  Thus, write as 

 

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(4.1)

 

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and to the equation 

 

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(4.2)

 

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add to both sides to get 

 

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(4.3)

 

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Square both sides to get 

 

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(4.4)

 

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then impose to get 

 

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(4.5)

 

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Bring everything to the left (most easily done with the Context Menu) to obtain 

 

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(4.6)

 

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then expand the parentheses.  Again, this is most easily done with the Context Menu, but the expand command will suffice. 

 

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(4.7)

 

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The route taken to shouldn't distract us from the next step where we attempt to factor this expression.  Again, we could have used the Context Menu, but the factor command will suffice. 

 

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The expression does not factor over the rationals.  The expression can be made more compact if we write it as 

 

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(4.8)

 

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Although the expression does not factor over the rationals, it does factor over the reals, as we see from 

 

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The coefficients in the factored form are approximations of exact representations of irrational numbers.  Maple can obtain exact expressions for the zeros of the quartic polynomial but these expressions are exceptionally unwieldy. Indeed, the solve command provides the solutions in the compact RootOf form shown below 

 

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but an application of the allvalues command to just the first such solution yields the cumbersome expression 

 

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The interested reader is welcome to applying allvalues to the other exact solutions as well. 

 

To obtain feedback on the calculations factor executes as it attempts to factor , execute 

 

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then 

 

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factor/polynom: polynomial factorization: number of terms 5 factor/unifactor: entering factor/unifactor: polynomial has degree 4 with 8 digit coefficients factor/linfacts: computing the linear factors factor/linfacts: there are 0 roots mod 29 factor/fac1mod: entering factor/fac1mod: found prime 7 factor/fac1mod: distinct degree factorization factor/fac1mod: degree set {0} factor/fac1mod: polynomial proven irreducible by degree analysis factor/unifactor: exiting

 

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In essence, Maple has proven that is irreducible over the rationals.  It would be a significant challenge for a student to solve the original trig equation completely "by hand" without some form of advanced technology.  Maple's Context Menu system, along with access to palettes and the "Equation Manipulator" make exploring and solving the given equation feasible. 

 

Because Maple stores the generated information in a "remember table," executing factor a second time will suppress the data, as we see from 

 

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It makes it seem as though "it doesn't work anymore."  However, the cure is 

 

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factor/polynom: polynomial factorization: number of terms 5 factor/unifactor: entering factor/unifactor: polynomial has degree 4 with 8 digit coefficients factor/linfacts: computing the linear factors factor/linfacts: there are 0 roots mod 29 factor/fac1mod: entering factor/fac1mod: found prime 7 factor/fac1mod: distinct degree factorization factor/fac1mod: degree set {0} factor/fac1mod: polynomial proven irreducible by degree analysis factor/unifactor: exiting

 

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