Lesson 1 George W. Ferris’ Day Off Develop Understanding

Jump Start

Find the following trigonometric ratios and angles for the given triangle:

1.

$\sin A =$

2.

$\cos A =$

3.

$\tan A =$

4.

$\sin B =$

5.

$\cos B =$

6.

$\tan B =$

7.

$m ∠ A =$

8.

$m ∠ B =$

Learning Focus

Apply right triangle trigonometry to a circular context.

How can right triangle trigonometry be applied to find how far points on a circle are away from a horizontal diameter of the circle?

Open Up the Math: Launch, Explore, Discuss

Perhaps you have enjoyed riding on a Ferris wheel at an amusement park. The Ferris wheel was invented by George Washington Ferris for the 1893 Chicago World’s Fair.

Carlos, Clarita, and their friends are celebrating the end of the school year at a local amusement park. Carlos has always been afraid of heights, but today he has chosen to ride on the amusement park Ferris wheel with his friends. As Carlos waits nervously in line, he has been able to gather some information about the wheel. By asking the ride operator, he found out that this wheel has a radius of $25 feet$ , and its center is $30 feet$ above the ground. With this information, Carlos is trying to figure out how high he will be at different positions on the wheel.

1.

How high above the ground will Carlos be when he is at the top of the wheel? (To make things easier, think of his location as a point on the circumference of the wheel’s circular path.)

2.

How high will he be when he is at the bottom of the wheel?

3.

How high will he be when he is at the positions farthest to the left or the right on the wheel?

Because the wheel has ten spokes, Carlos wonders if he can determine the height of the positions at the ends of each of the spokes as shown in the diagram. Carlos has just finished studying right triangle trigonometry and wonders if that knowledge can help him.

4.

Find the height of each of the points labeled $A - J$ on the Ferris wheel diagram below. Represent your work on the diagram so it is apparent to others how you have calculated the height at each point.

Height at $A$ =

Height at $B$ =

Height at $C$ =

Height at $D$ =

Height at $E$ =

Height at $F$ =

Height at $G$ =

Height at $H$ =

Height at $I$ =

Height at $J$ =

Pause and Reflect

5.

Describe a general procedure for finding how far a point on a circle is above or below the center of the circle.

Ready for More?

Describe a general strategy or formula for finding the vertical distance above the ground at any endpoint of a spoke of a Ferris wheel if the wheel has $12$ evenly spaced spokes instead of $10$ . Assume that one pair of spokes forms a horizontal line segment through the center of the wheel.

What facts about the wheel do you need to know to do this? Make up your own values for quantities you need to know.

Takeaways

The right triangles we have drawn in the circles today are called reference right triangles.

To find how far a point on a circle is located above or below the center of a circle:

A general rule for finding the height above the ground of a rider on Carlos’s Ferris wheel is:

Vocabulary

reference triangle
vertical height
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to place reference right triangles on a circle in order to find the distance a point on the circle is above or below the center of the circle. This is useful for finding the height above ground of a point on a circular object like a bicycle tire or a Ferris wheel.

Retrieval

1.

Find the other two trigonometric ratios based on the one that is given.

$\sin θ = \frac{\sqrt{5}}{3}$

a.

$\cos θ =$

b.

$\tan θ =$

2.

In problem a, divide out all common factors. In problem b, multiply. Divide out all common factors in your answer. (Assume no denominator equals $0$ .)

a.

$\frac{2 x - 6}{9 - 3 x}$

b.

$\frac{x^{2} - 5 x + 6}{3 x + 12} \cdot \frac{4 x + 16}{x^{2} + 3 x - 10}$