Lesson 2 “Sine” Language Solidify Understanding

Learning Focus

Write a trigonometric function to model a context.

How can I determine the vertical height of a rider on a moving Ferris wheel?

Open Up the Math: Launch, Explore, Discuss

In Lesson 1, you probably found Carlos’s height at different positions on the Ferris wheel using right triangles, as illustrated in the diagram.

Recall the following facts from the previous lesson:

The Ferris wheel has a radius of $25 feet$ .
The center of the Ferris wheel is $30 feet$ above the ground.

Carlos has also been carefully timing the rotation of the wheel and has observed the following additional fact:

The Ferris wheel makes one complete revolution counterclockwise every $20 seconds$ .

1.

How high will Carlos be $3 seconds$ after passing position $A$ on the diagram?

2.

Calculate the height of a rider at each of the following times $t$ , where $t$ represents the number of seconds since the rider passed position $A$ on the diagram. Keep track of any regularities you notice in the ways you calculate the height. As you calculate each height, plot the position on the Ferris wheel provided.

Elapsed time since passing position $A$	Calculations	Height of the rider
$1 second$
$2 seconds$
$3 seconds$
$6 seconds$
$7.5 seconds$
$8 seconds$
$14.5 seconds$
$18 seconds$
$23 seconds$
$28 seconds$
$36 seconds$
$37 seconds$
$40 seconds$

Pause and Reflect

3.

Examine your calculations for finding the height of the rider during the first $5 seconds$ after passing position $A$ (the first few values in the above table). During this time, the angle of rotation of the rider is somewhere between $0 °$ and $90 °$ . Write a general formula for finding the height of the rider during this time interval.

4.

How might you find the height of the rider in other “quadrants” of the Ferris wheel, when the angle of rotation is greater than $90 °$ ?

Ready for More?

It would be nice if our definition of sine would work for angles of rotation larger than the acute angles that can exist in a right triangle.

You may have noticed that a calculator is already designed to do so, giving us correct answers for the formula for the height of the rider as a function of elapsed time, even if the angle is larger than $90 °$ .

Create a list of conditions the definition of sine would have to meet in order to make a single equation for the height of the rider work in all quadrants of the Ferris wheel.

Takeaways

In a right triangle, the sine of an angle is defined as:

Angles of rotation are not restricted to acute angles, therefore, when working with angles of rotation:

Adding Notation, Vocabulary, and Conventions

Period of rotation:

Period of rotation for the Ferris wheel described in this task:

Angular speed:

Angular speed for the Ferris wheel described in this task:

Counterclockwise/Clockwise:

Direction of rotation for the Ferris wheel described in this task:

When describing the motion of the rider on the Ferris wheel, we will refer to the following quantities:

Elapsed time:

How to measure this quantity for the Ferris wheel described in this task:

Angle of rotation:

How to measure this quantity for the Ferris wheel described in this task:

Vocabulary

Lesson Summary

In this lesson, we found an equation for the vertical height of a rider on a moving Ferris wheel. That is, we treated the height of the rider as a function of the elapsed time since the rider passed the starting position, which we considered to be the farthest right position on the wheel.

Retrieval

1.

The graph of $f (x)$ is shown. Write the interval(s) where $f (x)$ is positive and the interval(s) where it is negative.

2.

Subtract. Divide out all common factors in your answers.

a.

$\frac{6 x + 5}{2 x + 6} - \frac{2 x - 7}{2 x + 6}$

b.

$\frac{8}{x + 5} - \frac{2 x - 15}{x^{2} - 25}$