Extend the process of inverse operations to trigonometric equations and inequalities.
Understand that using the unit circle will give infinite solutions to a trigonometric equation which may need to be restricted based on context and that an inverse trig function gives only one solution that may need to be expanded using symmetry.
Quick Lesson Plan
Today we’re revisiting the context of Lesson 6.7 where students constructed a model for the temperature outside Karissa’s home. Today students are given the model and have to find times when the temperature reaches a certain value. In essence, they are solving trig equations.
As always, we want to emphasize multiple representations and differing degrees of precision. Question 1 can be approached from an intuitive (temp starts at min, max must occur half-way through a cycle) or analytical (maximum value of F occurs when cos(πt/12)=-1) approach.
We now present an intuitive approach to solving equations that goes beyond just “doing the opposite”. First, students use the graph to estimate solutions, which then also allows them to check the reasonableness of their analytical solutions. Next, students think about the expression for F(t) in parts and isolate the impact of the variable t. In other words, if F(t)=75˚, what does the expression 10cos(πt/12) have to be so that when I subtract it from 80, I get 75? What, then, does the expression cos(πt/12) have to be? Question 3d may be tricky for students because they have to think of πt/12 as one quantity representing the angle. It can be helpful to use a picture or symbol like a little black box to represent πt/12. Overlaying this on the equation, students can first figure out what the “box” has to be, and then determine the value of t.
In question 3, the values are whole numbers and can be found using the unit circle. In questions 4-8, students must evaluate an inverse trig function using their calculator. Based on yesterday’s lesson, students should understand why the calculator gives only one value. In the answer key you will note that finding the other angle is a margin note, because students may or may not get there on their own. However, students should recognize that they are missing a solution since the graph gave them two intersection points. As you are debriefing the analytical answers, make sure to emphasize that students can check the reasonableness of their answers by comparing their solutions to their estimates in question 4. Since t=7.16 is very close to t=7 and t=16.84 is close to t=17, the solutions seem reasonable.
How many times will the temperature be exactly 83˚, according to this model?
Let’s pretend that the input is “box”. What does box have to be?
(Question 7) Is this a value you can find using your unit circle?
Why does your calculator give you only one value when we can see that there are two solutions?
Where else is cos(theta)=-0.3? How does having one solution help you find the other one?
Students should see in this lesson that solving a trigonometric equation is just like solving any other equation; it requires isolating the impact of the independent variable. We encourage you to avoid language like “move to the other side” and “get x by itself” as these can sometimes obscure the mathematical concept of undoing an equation and pinpointing the effect of one variable. We don’t want to focus on the physical location of the variable!
Because trigonometric functions are periodic, the number of solutions to any equation depends on the context. The unit circle provides infinite angles at which the sine, cosine, and tangent achieves a particular value (given that the value is in the range of the function). Students should not assume that they are only looking for solutions on the interval [0, 2π]! The inverse trig functions provide only one angle for a given ratio. But that doesn’t mean there is only one value that satisfies the equation. Questions will often specify an interval on which solutions should be found or the context will determine the interval. The best way for students to get good at this is to practice with varied examples!
Question 4b on the Check Your Understanding is challenging! There are 6 solutions (!) which students can find by realizing every 8 minutes Patty is in the exact same position, since that is the length of one full revolution of the wheel. Note that students are looking for all solutions on the interval [0,24]. This is an example of how the context of a problem can dictate the range of the solutions.