2.2 - Solutions to Trigonometric Equations

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Objectives:

• Represent solutions to an equation graphically.
• Understand the process of finding all $\theta$ satisfying $f(\theta)=y$, where $y$ is fixed and $f$ is a periodic function.
• Use the unit circle to solve equations of the above type for $f$ a trigonometric function.

Definitions

initial solutions

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Lesson Guide

Solutions as Intersection Points

Have students recall that solutions to mathematical equations can be viewed graphically as intersection points. Sketch a couple of examples, such as those shown below, with different numbers of solutions. In particular, point out (for these examples) that the quadratic equation yields two solutions, but the trigonometric function appears to have infinitely many (though not all solutions are visible).

Solutions to the equation $x^2-2=1$ are shown in the graph:

Solutions to the equation $\cos(\theta)=2/3$ are shown in the graph:

Solving Trigonometric Equations for All Solutions

Have students work through Problem 1 on this worksheet. Ask students what patterns they notice in their solutions to part (c); try to guide them to see that because the graph is periodic with period 4, they may express their solutions as $1+4k$ and $3+4k$ (for any positive integer choice of $k$) if the Ferris wheel makes infinitely many rotations. Build on this idea with an example such as the following:

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Solve the trigonometric equation $\cos(\theta)=\sqrt{3}/2$.

The key to solving this type of problem is to find the ``core" or ``initial" solutions to the equation - that is, the solutions over a single repeated segment. We can then translate them to find the remaining solutions. Consider the solutions to the equation as intersection points:

• $\cos(\theta) = \sqrt{3} / 2$ holds true for two angles on the unit circle: $\theta = \pi / 6$ corresponding to Solution $A$ in the diagram above and $\theta = 11\pi/6$, corresponding to Solution $B$.
  • These are solutions appearing on the unit circle -- remind students to check their unit circles for exact solutions first when solving equations of this type.
• Since $\cos(\theta)$ is periodic, we can take our two initial solutions of $\theta= \pi/6$ and $\theta= 11\pi/6$, and add or subtract multiples of $2\pi$ in order to find additional solutions. For example, two additional solutions are given by: $\theta = \pi/6 - 2\pi$ and $\theta= 11\pi/6 - 2\pi$.
• To account for all solutions, we write: $\theta=\pi/6 + 2\pi k$ and $\theta=11\pi/6 + 2\pi k.$
  • Emphasize that the ``$k$" here can be any integer. Show them the solutions corresponding to $k=0,-1$, and $1$ on the graph.

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Have students work through Problem 2 on their worksheets.

Discuss the case in which the initial solutions are not angles on their unit circles. You can make up your own example here, or do Problem 3(a) as a class. Remind students that they can check their solutions, and walk them through the process of doing so in this scenario.

Make sure that students realize that it will not always be the case that we find exactly two initial solutions (there may be more or fewer) or that we always add multiples of \emph{$2\pi$} (the period of the function may be different) -- this will be explored in Problems 4 and 5 of this worksheet. Consider going over these problems as a class.

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