2.3 - Applications of Inverse Trigonometric Functions (Parts I & II)

Prior Lesson | Next Lesson

Objectives:

• Solve more complicated trigonometric equations.
• Solve trigonometric equations on a given interval.
• Use factoring to solve equations containing multiple trigonometric functions.

---

Lesson Guide

There are two worksheets for this section. The second worksheet contains no new material, but is simply extra practice.

Solving Trigonometric Equations

We have previously seen how to solve basic trigonometric equations involving functions of just $\theta$, for example $\sin(\theta)=1/2$. Today we will consider more complex examples. Start off with an example such as the one below.

---

Solve the trigonometric equation $2\sin(3\theta-1)=-1$.

Point out that key difference between this example and our previous ones is that the input into sine is $3\theta-1$ rather than just $\theta$.

• First, we isolate the sine function: $\sin(3\theta-1)=-1/2$.
• Next, we find the solutions to this equation: since $-1/2$ corresponds to a standard angle on the unit circle, the initial solutions satisfy: $3\theta-1=7\pi/6$ and $3\theta-1=11\pi/6$.
  • The quantity inside the trigonometric function, $3\theta-1$, is the input to the sine function. We set the \textit{entire input} equal to $7\pi/6$ and $11\pi/6$ to determine the initial solutions.
• Using these initial solutions, we find \textit{all possible solutions} for $3\theta-1$ by adding copies of the period of sine: $3\theta-1=7\pi/6 + 2\pi k$ and $3\theta-1=11\pi/6+2\pi k$, where $k$ is any integer. Notice we do this before solving for $\theta$.
• Solving for $\theta$ produces our final solutions of:
  • $\theta=\frac{7\pi/6 +2\pi k+1}{3}=\frac{7\pi +6}{18}+\frac{2\pi}{3} k \approx 1.56 +\frac{2\pi}{3} k$, and
  • $\theta=\frac{11\pi/6 +2\pi k+1}{3}=\frac{11\pi +6}{18}+\frac{2\pi}{3} k \approx 2.25 +\frac{2\pi}{3} k$.

Point out that in the example, 1.56 and 2.25 represent the initial solutions for $\theta$ (not $3\theta-1$), and the period of $2\sin(3\theta-1)$ is equal to $2\pi/3$. When teaching this process, you can instead solve for the initial solutions of 1.56 and 2.25, and then add copies of $2\pi/3$. However, the method taken above, in which copies of $2\pi$ are added \emph{before} solving for $\theta$, may be more straightforward for students.

---

Have students work through Problems 1(a) and 1(b) on the Part I worksheet.

Solutions on a Given Interval

We know how to find \textit{all possible} solutions to many trigonometric equations now, but sometimes we want to know the solutions to an equation \textit{only on an interval.} Illustrate this using an example such as the one below:

---

Find all solutions to $2\sin(3\theta - 1) = -1$ in the interval $[-\pi/2,3\pi / 2]$.

• We've found that all of the solutions to the equation $2\sin(3\theta - 1) = -1$ are of form $1.56 + \frac{2\pi}{3}k$ or $2.25 + \frac{2\pi}{3}k$ for some integer $k$.
• Now we test values of $k$ to figure out which solutions lie within the interval $[-\pi/2,\,3\pi/2]$:
  • $k=-2$: $\theta=1.56+\frac{2\pi}{3}\cdot (-2)\approx -2.63$ and $\theta=2.25+\frac{2\pi}{3}\cdot (-2)\approx -1.94$
  • $k=-1$: $\theta\approx -0.54$ and $\theta\approx 0.16$
  • $k=0$: $\theta\approx 1.56$ and $\theta\approx 2.25$
  • $k=1$: $\theta\approx 3.65$ and $\theta\approx 4.35$
  • $k=2$: $\theta\approx 5.75$ and $\theta\approx 6.44$
• Since $[-\pi/2,\,3\pi/2]$ is approximately $[-1.57,4.71]$, we can cross off all solution sets except $k=-1$, $k=0$, and $k=1$.

We can see these solutions in the graph below:

---

Have students work through Problems 1(c) and 2 on the Part I worksheet.

This section also includes solving equations that have more than one occurrence of sine or cosine, in which students may need to factor while solving. Give an example (for example, ask how they might solve $\sin(\theta)\tan(\theta) - \sqrt{3}\sin(\theta)=0$ for all values of $\theta$).

Have students complete the worksheet.

Comments

---

In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.