Trigonometry Notes: Solving Trig Equations

This material appears in sections 5.3.

• Equations that are linear in the trig functions:
  

  The strategy is:

  

  • 
    Find all solutions to:

    

    There are two angles between 0 and that will give you sine = . They are the two angles where the horizontal line at height meets the unit circle. One of these angles is , the other is .

    

    This gives us all the solutions that live in one cycle. This function has period so we should also include solutions that differ from these two by integer multiples of .

    So all solution are of the form:

    

    or

    

  • 
    Find any solution to

    

  • 
    Find the smallest positive solution to

    

  • 
    Find all solutions to

    Tangent doesn't repeat values within a cycle, so arctan will find the only solution in a cycle, and then we can extend by periodicity.

    

    So one solution is , and since tangent is -periodic, all solutions are of the form:

    

  • 
    Find all solutions to:

    

    There are two angles between and that have cosine=. They are and .

    So

    

    Finally, the preiod is , so we get all solutions by adding multiples of the period to the two particular solutions. So all solutions are of the form: