Lesson 9 Water Wheels and the Unit Circle Practice Understanding

Jump Start

Quick graphs again: In the previous lesson, you learned how to graph sine and cosine functions using the real number values along the $x$ -axis to represent the angle of rotation. Of course, in our Ferris wheel model, we still want the $x$ -axis to represent time in seconds. And now we have the possibility to describe the angular speed of the Ferris wheel in radians. Work with a partner to sketch a graph of each of the following functions on an appropriate grid. You need to show one complete period, so choose your grid wisely. (Hint: Can you find the one that looks most like the Ferris wheel equations?)

Use the quick-graph strategy: plot points along the midline to define a period, and plot the maximum and minimum points to define the range.

1.

$f (x) = 2 \cos (\frac{1}{2} x) + 4$

2.

$g (t) = 2 \sin \frac{30 °}{seconds} \cdot t + 2$

3.

$h (t) = 4 \sin (π t)$

Learning Focus

Apply special right triangles to the unit circle.

Are there any angles for which I can find the value of the sine or cosine without using a calculator?

Open Up the Math: Launch, Explore, Discuss

Water wheels were used to power flour mills before electricity was available to run the machinery. The water wheel turned as a stream of water pushed against the paddles of the wheel. Consequently, unlike Ferris wheels that have their centers above the ground, the center of the water wheel might be placed at ground level, so the lower half of the wheel would be immersed in the stream.

1.

The following diagrams show potential designs for a water wheel. Each of the $12 spokes$ of the water wheel will measure $1 meter$ . In addition to the spokes, the designer wants to add braces to provide additional strength. Two potential placements for the braces are shown in the following diagrams. (The braces and the spoke to which they are attached form a right angle.)

Find the measures of $∠ B A C$ and $∠ A B C$ in each diagram.

Find the exact lengths of $\overset{―}{A B}$ , $\overset{―}{A C}$ , and $\overset{―}{B C}$ . Explain how you found these lengths exactly.

Label the exact coordinates of point $B$ in each diagram.

2.

Based on your work in problem 1, label the exact values of the $x$ - and $y$ -coordinates for each point on the drawing of the water wheel. Remember that the center of the wheel is at ground level, so points below the center of the wheel should be labeled with negative values. As in the Ferris wheel models, label points to the left of the center with negative coordinates also.

3.

Use the diagram in problem 2 to give exact values for the following trigonometric expressions.

a.

$\sin (\frac{π}{6}) =$

b.

$\sin (\frac{5 π}{6}) =$

c.

$\cos (\frac{7 π}{6}) =$

d.

$\sin (\frac{π}{3}) =$

e.

$\cos (\frac{π}{6}) =$

f.

$\cos (\frac{11 π}{6}) =$

g.

$\sin (\frac{3 π}{2}) =$

h.

$\cos (π) =$

i.

$\sin (\frac{7 π}{3}) =$

4.

Here is a plan for an alternative water wheel with only $8 spokes$ . Label the exact values of the $x$ - and $y$ -coordinates for each point on the following schematic drawing of the water wheel. (Hint: You might want to begin this work by finding the length of the “brace” shown in the diagram.)

5.

Use the diagram in problem 4 to give exact values for the following trigonometric expressions.

a.

$\sin (\frac{π}{4}) =$

b.

$\sin (\frac{5 π}{4}) =$

c.

$\cos (\frac{3 π}{4}) =$

d.

$\cos (\frac{π}{4}) =$

e.

$\cos (- \frac{π}{4}) =$

f.

$\sin (\frac{7 π}{4}) =$

g.

$\sin (\frac{3 π}{2}) =$

h.

$\cos (\frac{3 π}{2}) =$

i.

$\sin (\frac{11 π}{4}) =$

Pause and Reflect

During the spring runoff of melting snow, the stream of water powering this water wheel causes it to make one complete revolution counterclockwise every $3 seconds$ .

6.

Write an equation to represent the height of a particular paddle of the water wheel above or below the water level at any time $t$ after the paddle emerges from the water.

Write your equation so the height of the paddle will be graphed correctly on a calculator set in degree mode.

Revise your equation so the height of the paddle will be graphed correctly on a calculator set in radian mode.

During the summer months, the stream of water powering this water wheel becomes a “lazy river” causing the wheel to make one complete revolution counterclockwise every $12$ seconds.

7.

Write an equation to represent the height of a particular paddle of the water wheel above or below the water level at any time $t$ after the paddle emerges from the water.

Write your equation so the height of the paddle will be graphed correctly on a calculator set in degree mode.

Revise your equation so the height of the paddle will be graphed correctly on a calculator set in radian mode.

Ready for More?

Suppose you know the sine of an angle is $- 0.425$ . What are possible values for the cosine of the angle? How can you use the unit circle to help you think about this question?

Takeaways

Because of the relationships found in special right triangles (see diagrams), the coordinates of the points on the unit circle for angles of rotation that are multiples of $\frac{π}{4}$ (or $45 °$ ), $\frac{π}{6}$ (or $30 °$ ), and $\frac{π}{3}$ (or $60 °$ ) can be labeled with exact values, such as shown for the first quadrant of the unit circle diagram.

The labeled unit circle is like a trigonometry table for finding trigonometry values for these special angles.

For example, to find $\sin (\frac{5 π}{3})$ , I would:

When graphing trigonometric functions on my calculator that represent contexts that involve angles of rotation, I can determine whether to use degree or radian mode by:

When graphing trigonometric functions on my calculator that represent contexts for which the real numbers are the domain, I should:

Vocabulary

special right triangles
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned that the values of some trigonometric expressions can be found exactly, instead of as decimal approximations. This occurs because we can find the exact side lengths for special right triangles with a hypotenuse of $1 unit$ . We can then use these lengths to label the coordinates of points around the unit circle that can be identified by placing these right triangles in various positions where they fit within the unit circle.

Retrieval

1.

Find a negative angle of rotation that is coterminal with $θ = 120 °$ .

Sketch and label both angles in standard position.

2.

The number of degrees an object passes through during a given amount of time is called angular speed. For instance, the second hand on a clock has an angular speed of $\frac{360 °}{min}$ , while the minute hand on a clock has an angular speed of $\frac{360 °}{hr}$ . (Recall that a revolution is a full circle or $360 °$ .)

a.

What is the angular speed of the second hand on a clock in degrees per second?

b.

What is the angular speed of the minute hand on a clock in degrees per second?

c.

What is the angular speed of the hour hand in degrees per hour?