Lesson 16

This warm-up prompts students to compare four trigonometric functions. It gives students a reason to use language precisely (MP6) and gives the opportunity to hear how they use terminology and talk about characteristics of the items in comparison to one another.

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn't belong.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as scale factor, horizontal translation, or period. Also, press students on unsubstantiated claims.

This activity continues to develop the idea of period for a trigonometric function both graphically and from the point of view of expressions and equations. In the previous lesson, students first saw a trigonometric function whose period was scaled. The function \(\cos(2\theta)\) has period \(\pi\) because the expression \(2\theta\) goes through all values between 0 and \(2\pi\) when \(\theta\) goes from 0 to \(\pi\). For the same reason, \(\sin(5\theta)\) has period \(\frac{2\pi}{5}\). From a graphical point of view, decreasing the period means that the wave is compressed horizontally.

The new content in this activity is that students begin to look at functions defined by expressions like \(\cos(\pi \theta)\). The impact of the coefficient \(\pi\) is again to change the period, here from \(2\pi\) to 2, since \(\frac{2\pi}{\pi} = 2\). Finding trigonometric functions whose period is a whole or rational number is important in the applications that students study for the remainder of this unit. When they model real world phenomena with trigonometric functions, the period of those phenomena is usually a rational number.

As students calculate the period for the 5 functions listed, some may take the coefficient of \(\theta\) as the period. For example, they may think that the period of \(y=\cos(10\theta)\) is 10. Ask, “What value does \(\theta \) need to be in order for \(10\theta\) to be 0? To be \(2\pi\)?” (0 and \(\frac{2\pi}{10}\), respectively.) “What does that tell you about the period of \(\cos(10\theta)\)?” (That it repeats every \(\frac{2\pi}{10}\), so that's the period of \(y=\cos(10\theta)\).)

Focus student attention on the two ways of understanding the period of a trigonometric function that appear in this activity: interpreting a graph and interpreting an expression. For the graphical interpretation, ask students questions like:

• "How can you find the period of a trigonometric function from its graph?" (The period is how long it takes the function to complete one full "wave" or cycle of output values.)
• "What is the period of the trigonometric function from the last problem? Explain how you know." (1, because the graph completes one cycle when the input increases by 1.)

For interpreting expressions, ask students questions like:

• “When \(\theta = 0\), what value does \(2\pi \theta\) take? What about \(\cos(2\pi \theta)\)?” (0 and 1.)
• “When \(\theta = 1\) what value does \(2\pi \theta\) take? What about \(\cos(2\pi \theta)\)?” (\(2\pi\) and 1.)
• “How long does it take the graph of \(y=\cos(2\pi \theta)\) to complete one full period? Explain how you know.” (1 because whenever \(\theta\) increases by 1, \(2\pi \theta\) goes through all values between 0 and \(2\pi\).)

Highlight that the function \(y = \cos(2\pi \theta)\) defines the graph in the last question because the period is 1. The function \(y=\cos(4\pi \theta)\) defines the third graph of the first problem as its period is \(\frac{1}{2}\).

In this activity, students return to the context of a Ferris wheel with their new understanding of how to transform a periodic function by changing the midline, amplitude, and period. The purpose of this activity is for students to interpret a sine function in context and then make a sketch of the function.

Monitor for students using clear reasoning about the expression for \(f\) as they identify the diameter, time per revolution, and times the passenger seat is at its lowest point. For example, students may make and solve equations like \(\frac{2\pi t}{30}=2\pi\) in order to determine the time \(t\) for one full revolution.

A note on the use of \(t\) as a variable: Until this activity, the input for trigonometric functions has typically been \(\theta\). In a previous course students used \(\theta\) to represent angles in different geometric shapes, so its use throughout this unit was meant to remind students of the connection between cosine and sine and the unit circle. Now, however, students are transitioning to working with functions modeling contexts where the input isn't an angle, so it makes sense to choose a variable representative of the input value. Since \(f\) takes an input of time and gives and output of height, the variables \(t\) and \(h\) are reasonable choices. If time allows and students wonder why \(t\) is used instead of \(\theta\), invite students to suggest why they think the input variable is \(t\) and highlight the reasons listed here for the change.

Tell students that they will be examining the position of a seat on a very large Ferris wheel. The Ferris wheel moves slowly enough that passengers can get on and off while the Ferris wheel continues its motion.

It time allows, recommend students try making a sketch of \(f\) before they graph the function using technology.

If students are unsure how to interpret the expression inside the sine function, ask them to consider what value they would use for \(t\) if they want the input expression to be 0, \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), or \(2\pi\).

Select previously identified students to share their reasoning about the first three questions. Highlight reasoning such as:

• 260 feet is the amplitude. The value of \(260\sin(\frac{2\pi t}{30})\) will go between -260 and +260 for different inputs. This means the diameter of the wheel is 520 feet.
• The expression \(\frac{2\pi t}{30}\) gives the period. When \(t\) goes from 0 to 30, this expression goes from 0 to \(2\pi\) so the Ferris wheel makes one full revolution in 30 minutes meaning its period is 30 minutes.
• The function has its lowest value when \(260\sin(\frac{2\pi t}{30})=\text-260\). This is when \(\sin(\frac{2\pi t}{30})=\text-1\), which happens when the input to sine is \(\frac{3\pi}{2}\). We can calculate the value of \(t\) by solving \(\frac{2\pi t}{30}=\frac{3\pi}{2}\), which is when \(t=22.5\).
• Since the period of the wheel is 30 minutes, then 22.5 minutes after 8 a.m. and every 30 minutes after that passenger seat is at its lowest point.

As students share, highlight how each of these features can be seen on a displayed graph of \(y=f(t)\).