# 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

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.

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## 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.

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The following diagrams show potential designs for a water wheel. Each of the

Find the measures of

and in each diagram.

Find the exact lengths of

, , and . Explain how you found these lengths exactly.

Label the exact coordinates of point

in each diagram.

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Based on your work in problem 1, label the exact values of the

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

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Here is a plan for an alternative water wheel with only

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

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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

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Write an equation to represent the height of a particular paddle of the water wheel above or below the water level at any time

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

### 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

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

## 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

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

For example, to find

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.

Find a negative angle of rotation that is coterminal with

Sketch and label both angles in standard position.

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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

#### a.

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

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

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