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
1.
Write the equation of the height of the rider at any time
2.
At what time(s) is the rider
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
5.
For the equation you wrote in problem 4, at what time(s) is the rider
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
6.
Write the functions that model the frequency of vibration and loudness for the following notes:
a.
Middle C, played on a violin at
b.
The A note used to tune the piano at
Ready for More?
Choose any other starting position and write the equation of the height of the rider at any time
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.
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
2.
Multiply.
a.
b.
3.
Find the quadratic equation with solutions