Trigonometric Equations

    A trigonometric equation that is true for only some values of the variable but not for others is called a conditional trigonometric equation. In this topic we give examples of solving different types of trigonometric equations. Some trigonometric equations can be solved by factoring and others by using trigonometric identities. One technique is to square both sides which used be used with caution. In this case, checking for extraneous solutions is not optionally.

Example Trigonometric Equations) Solve the trigonometric equation over the interval [ ).

    Solution. Take the square root of both sides we have,








Example Trigonometric Equations) Solve the trigonometric equation over the interval [ ).

    Solution.  Take the square root of both sides we have,







    
Example Trigonometric Equations) Solve the trigonometric equation over the interval [ ).

    Solution. We have,
    











    
Example Trigonometric Equations) Solve the trigonometric equation over the interval [ ).

    Solution. We have,

Example (Trigonometric Equations) Find all solutions to the trigonometric equation

    Solution. Let be any integer. We have,
    













Example (Trigonometric Equations) Find all solutions to the trigonometric equation
    
    Solution. Let be any integer. We have,














Example (Trigonometric Equations) Find all solutions to the trigonometric equation

    Solution. Let be any integer. We have,  
















Example (Trigonometric Equations) Find all solutions to the trigonometric equation

    Solution. Let be any integer. Squaring both sides we have,















    

Example (Solving Trigonometric Equation Numerically) Approximate the solutions to the trigonometric equation on the interval

    Solution. From the initial graph, we conclude there are four solutions on


Zooming in on the first two we have,  



After zooming in more, we find that and Thus the four solutions are






Example (Solving Trigonometric Equation Numerically) Approximate the solutions to the trigonometric equation on the interval

    Solution. From the initial graph, we conclude there are two solutions on


Zooming in on the first solution we find that



Zooming in on the second solution we find that

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Trigonometric Equations
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/trigonometric-equations.html