Objectives
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Give complete solutions to elementary trigonometric equations.
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Understand elementary trigonometric equations geometrically via a unit circle picture.
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Solve more complicated trigonometric equations by reducing to elementary ones.
Section 2.19 Solving trigonometric equations
By an elementary trigonometric equation, we will mean an equation of one of the three forms,
\begin{align*}
\cos\theta \amp =a \amp \sin\theta\amp =b \amp \tan\theta\amp=c\text{,}
\end{align*}
where \(a\text{,}\) \(b\text{,}\) and \(c\) are fixed constants, and \(\theta\) is the unknown. As a result of the periodic nature of trigonometric functions, combined with some intrinsic symmetries of the unit circle, we will see that elementary trigonometric equations will usually have infinitely many solutions; and indeed, the solutions often come in two independent infinite lists!
Let’s begin with the equation form \(\cos\theta=a\text{.}\) First observe that since \(\range \cos=[-1,1]\text{,}\) this equation has a solution in the unknown \(\theta\) if and only if \(-1\leq a\leq 1\text{.}\) Next, using the definition of \(\cos\theta\) as the \(x\)-coordinate of the corresponding point \(P_\theta\) on the unit circle, we see that a solution \(\theta\) corresponds to an angle \(\theta\) whose corresponding point \(P_\theta\) has \(x\)-coordinate equal to \(a\text{.}\) In other words, solutions to \(\cos\theta=a\) correspond to points on the intersection of the vertical line \(x=a\) and the unit circle \(x^2+y^2=1\text{.}\) As illustrated in Figure 2.19.1 the vertical line \(x=a\) will intersect the unit circle \(C\) either not at all (if \(a\notin [-1,1]\)), in exactly one point (if \(a=\pm 1\)), or in exactly two points (if \(a\in (-1,1)\)).
Let’s see how to solve the equation \(\cos\theta=a\) in the case when \(a\in (-1,1)\text{.}\) An angle \(\theta\) solving this equation will have associated point \(P_\theta\) lying on the intersection of the line \(x=a\) and the unit circle. From the diagram above, we see there are two such points, and by symmetry, we see that if \(P_\theta\) is one of the points, then \(P_{-\theta}\) is the other, as indicated in Figure 2.19.2
Now, imagine we have found one solution \(\theta\) to \(\cos\theta=a\text{,}\) corresponding to the point \(P_\theta\text{.}\) Any angle coterminal with \(\theta\) will intersect the unit circle in the same point \(P_\theta\text{.}\) We conclude that any angle of the form \(\theta+2\pi n\text{,}\) \(n\in\Z\text{,}\) is also a solution to \(\cos\theta=a\text{.}\) The same reasoning tells us that any angle of the form \(-\theta+2\pi n\text{,}\) \(n\in \Z\) is a solution to \(\cos\theta=a\text{.}\) From a single solution \(\theta\) to the equation, we have thus generated two independent infinite lists of solutions:
\begin{align*}
\alpha \amp =\theta+2\pi n, \ n\in \Z\\
\beta \amp = -\theta+2\pi n, \ n \in \Z\text{.}
\end{align*}
Similar reasoning applies to an elementary sine equation \(\sin\theta=b\text{.}\) Now we are looking for angles \(\theta\) for which the corresponding point \(P_\theta\) has \(y\)-coordinate equal to \(b\text{.}\) This is a point lying on the intersection of the horizontal line \(y=b\) and the unit circle. For the case when \(b\in (-1,1)\text{,}\) if \(P_\theta\) is one such point, \(P_{\pi-\theta}\) is the other.
From the unit circle picture, we see that from a single solution \(\theta\) to \(\sin\theta=b\text{,}\) we can generate two independent infinite lists of solutions:
\begin{align*}
\alpha \amp =\theta+2\pi n, \ n\in \Z\\
\beta \amp = (\pi-\theta)+2\pi n, \ n \in \Z\text{.}
\end{align*}
Finally, consider the the equation \(\tan \theta=c\text{.}\) Recall that \(\tan\theta\) is the slope of the terminal edge of the angle \(\theta\text{.}\) As indicated below the angles \(\theta\) and \(\theta+\pi\) have terminal edges with the same slope. It follows that if \(\theta\) is one solution to \(\tan\theta=c\text{,}\) then \(\theta+\pi n\text{,}\) \(n\in \Z\) is the set of all solutions.
We summarize our conclusions with a general procedure for finding all solutions to an elementary trigonometric equation.
Procedure 2.19.5. Solving elementary trigonometric equations.
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Solving \(\cos\theta=a\).The equations\begin{align*} \cos\theta \amp =1 \amp \cos\theta\amp=-1 \end{align*}have solutions\begin{align*} \theta \amp =2\pi n,\ n\in \Z \amp \theta=\pi+2\pi n,\ n\in \Z\text{.} \end{align*}If \(a\in (-1,1)\) and \(\theta_0\) is one solution to \(\cos\theta=a\text{,}\) then the complete list of solutions is\begin{align} \theta \amp = \theta_0+2\pi n, \ n\in \Z \tag{2.88}\\ \theta \amp= -\theta_0+2\pi n, \ n\in \Z \text{.}\tag{2.89} \end{align}
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Solving \(\sin\theta=b\).The equations\begin{align*} \sin\theta \amp =1 \amp \sin\theta\amp=-1 \end{align*}have solutions\begin{align*} \theta \amp =\pi/2+2\pi n, \ n\in \Z \amp \theta=-\pi+2\pi n,\ n\in \Z\text{.} \end{align*}If \(b\in (-1,1)\) and \(\theta_0\) is one solution to \(\sin\theta=b\text{,}\) then the complete list of solutions is\begin{align} \theta \amp = \theta_0+2\pi n, n\in \Z \tag{2.90}\\ \theta \amp= (\pi-\theta_0)+2\pi n, \ n\in \Z \text{.}\tag{2.91} \end{align}
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Solving \(\tan\theta=c\).If \(\theta_0\) is a solution to \(\tan\theta=c\text{,}\) then the complete list of solutions is\begin{align*} \theta \amp = \theta_0+\pi n, \ n\in \Z \end{align*}
Note that except for the simple equations \(\cos\theta=\pm 1\) and \(\sin\theta=\pm 1\text{,}\) Procedure 2.19.5 does not give an explicit formula for the general solution to an elementary trigonometric equation: it only describes how to get all solutions starting from a given solution \(\theta_0\text{.}\) When solving elementary trigonometric equations by hand, the way to proceed is to find one solution \(\theta_0\) corresponding to one of the 16 familiar angles, and then write all other solutions in terms of \(\theta_0\text{.}\)
Example 2.19.6. Elementary trigonometric equations.
Find all solutions to the given equation.
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\(\displaystyle \cos\theta=-1/2\)
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\(\displaystyle \sin x=-\sqrt{2}/2\)
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\(\displaystyle \tan t=\sqrt{3}\)
Solution.
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First we observe that \(\theta_0=2\pi/3\) is one solution to the given equation, making use of the familiar angles in Figure 2.17.13. It follows that the complete list of solutions is\begin{align} \theta \amp = 2\pi/3+2\pi n, \ n\in \Z \tag{2.92}\\ \theta \amp= -2\pi/3+2\pi n, \ n\in \Z \text{.}\tag{2.93} \end{align}
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We observe that \(x_0=5\pi/4\) is one solution to the given equation, making use of the familiar angles in Figure 2.17.13. It follows that the complete list of solutions is\begin{align} x \amp = 5\pi/4+2\pi n, \ n\in \Z \tag{2.94}\\ x \amp= (\pi-5\pi/4)+2\pi n=(-\pi/4)+2\pi n, \ n\in \Z \text{.}\tag{2.95} \end{align}
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To find a familiar solution to \(\tan t=\sqrt{3}\text{,}\) it helps to write \(\sqrt{3}\) as \((\sqrt{3}/2)/(1/2)\text{,}\) since now we recognize the numerator as \(\sin \pi/3\) and the denominator as \(\cos\pi/3\text{.}\) It follows that \(t_0=\pi/3\) is a solution to the given equation, and thus that the complete list of solutions is\begin{align*} t \amp = \pi/3+\pi n, \ n\in \Z \text{.} \end{align*}
