Lesson 3 Getting on the Right Wavelength Practice Understanding

Learning Focus

Write equivalent sine and cosine equations.

Find the complete set of solutions for a trigonometric equation.

Model periodic contexts.

How do I write a cosine equation that is equivalent to a given sine equation and vice versa? Why might I want to do so?

How can I determine the times when a rider will be at certain positions on the Ferris wheel?

Open Up the Math: Launch, Explore, Discuss

The Ferris wheel in the diagram has a radius of $40 feet$ , its center is $50 feet$ from the ground, and it makes one revolution counterclockwise every $18 seconds$ .

1.

Write the equation of the height of the rider at any time $t$ , if at $t = 0$ the rider is at position $A$ (Use radians to measure the angle of rotation).

2.

At what time(s) is the rider $70 feet$ above the ground? Show the details of how you answered this problem.

3.

If you used a sine function in problem 1, revise your equation to model the same motion with a cosine function. If you used a cosine function, revise your equation to model the motion with a sine function.

4.

Write the equation of the height of the rider at any time $t$ , if at $t = 0$ the rider is at position $D$ .

5.

For the equation you wrote in problem 4, at what time(s) is the rider $80$ feet above the ground? Show or explain the details of how you answered this problem.

Pause and Reflect

We hear musical notes when a vibrating object, such as a violin string, causes our eardrum to vibrate at a specific frequency. For example, we hear the note referred to as “middle C” on a piano when the eardrum vibrates approximately $260$ times per second, and pianos are tuned so that the A note above middle C causes the eardrum to vibrate $440$ times per second. Because this vibration is periodic, it can be modeled with trigonometric functions. The amplitude of a sound wave determines its loudness, which is measured in decibels. A standard piano produces soundwaves with a loudness between $60$ and $70$ decibels, and a violin produces soundwaves between $82$ and $92$ decibels.

6.

Write the functions that model the frequency of vibration and loudness for the following notes:

a.

Middle C, played on a violin at $90$ decibels.

b.

The A note used to tune the piano at $60$ decibels.

Ready for More?

Choose any other starting position and write the equation of the height of the rider at any time $t$ , if at $t = 0$ the rider is at the position you chose. (Use radians to measure the angle of rotation.) Also change other features of the Ferris wheel, such as the height of the center, the radius, the direction of rotation, and/or the length of time for a single rotation. (Record your equation and description of your Ferris wheel here.)

Trade the equation you wrote with a partner and see if they can determine the essential features of your Ferris wheel: height of center, radius, period of revolution, direction of revolution, starting position of the rider. Resolve any issues where you and your partner have differences in your descriptions of the Ferris wheel modeled by your equations.

Takeaways

Every sine function can be represented by

Both the sine and cosine functions will have the same

While only one solution to a trigonometric equation can be found using , the solution set of every trigonometric equation .

We can visualize all of the solutions to a trigonometric equation by

Periodic behavior is often described in terms of frequency,

The frequency of a trigonometric function is

Adding Notation, Vocabulary, and Conventions

Because trigonometric functions are periodic, trigonometric equations have

To list the complete solution set:

Vocabulary

frequency
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we reviewed writing trigonometric functions and solving trigonometric equations to model situations in a context. We observed that equivalent sine and cosine functions can be written to model the same context, and that equivalent forms of equations that represent a horizontal translation of a trigonometric function emphasize changing different quantities in the context, such as the initial position or the start time.

Retrieval

1.

Use what you know about the values of sine and cosine on the unit circle, and the definition of tangent in a right triangle, to find the value of tangent $θ$ .

2.

Multiply.

a.

$(2 + 7 i) (5 - 4 i)$

b.

$(2 + 10 i) (2 - 10 i)$

3.

Find the quadratic equation with solutions $x = (2 + 10 i)$ and $x = (2 - 10 i)$ . Assume the coefficient of $x^{2}$ is $1$ .