Lesson 5 Rays and Radians Solidify Understanding

Jump Start

Madison and her friends are trying to write a good definition of radians for their Exit Ticket. Here are some of their responses. Decide if each statement is true or false. Then decide if each statement is useful as written or requires more precision.

Student	A radian is...	True or False?	Useful or Lacks Precision?
Travis	A new way of measuring angles.
Anushka	A way of measuring a central angle using arc length, rather than degrees.
Aaliyah	The measure of an arc whose length equals the radius of the circle.
Aryan	The ratio of arc length to radius.
Mateo	A different unit for measuring angles rather than using degrees.

Learning Focus

Measure angles in radians.

How do we convert between degree and radian measurement?

How can I visualize the size of an angle when its measure is given in radians instead of degrees?

Open Up the Math: Launch, Explore, Discuss

In the previous task, Madison’s Round Garden, Madison finds a new way to measure angles. Apparently Madison is not the first person to have this idea for measuring an angle in terms of arc length, but once she is aware of it, she decides to examine it further.

Here are some of Madison’s questions. See if you can answer them.

1.

Since a $40 °$ angle measures $0.698 radians$ (to the nearest thousandth), a $50 °$ angle measures $0.873 radians$ , and a $60 °$ angle measures $1.047 radians$ , what angle, measured in degrees, measures $1.000 radians$ ?

2.

A circle measures $360 °$ . How many radians is that?

3.

The formula Madison has been using to calculate radian measurement for an angle that measures $n °$ on a circle of radius $r$ is $\frac{\frac{n °}{360 °} (2 π r)}{r} = x radians$ .

Is there a simpler formula for converting degree measurement to radian measurement?

4.

What formula might you use to convert radian measurement back to degrees?

Pause and Reflect

Madison is so excited about radian measurement that she decides to learn more about it by going online. She finds this statement: An arc of a circle with the same length as the radius of that circle corresponds to an angle of $1 radians$ . A full circle corresponds to an angle of $2 π radians$ .

5.

Why is the first sentence in this statement true?

6.

Why is the second sentence in this statement true?

7.

Use a piece of string and a circle to determine how many radians there are in a full circle. Does this experiment verify or contradict the statements Madison finds online?

Madison finds this idea of writing radian measurement in terms of $π$ appealing. Since a circle measures $2 π radians$ , she reasons that half of a circle, $180 °$ , would measure $π radians$ ; and that a quarter of a turn, a right angle, would measure $\frac{π}{2}$ radians. Suddenly Madison realizes that while she has been deep in thought thinking about this new idea, she has been fiddling with her protractor. Now her attention focuses on this tool for measuring angles.

Like Madison, you have probably used a protractor to measure angles. A protractor is usually marked to measure angles in degrees. Madison decides she would like to create a protractor to measure angles in radians.

8.

Label the protractor in radians, using fractions involving $π .$ You should label every $10 °$ from $0 °$ to $180 °$ . For example, rays passing through the $0 °$ and $40 °$ angle mark would form an angle measuring $\frac{4}{18} π$ (or $\frac{4 π}{18} = \frac{2 π}{9}$ ) radians, so we would label the tic mark at $40 °$ as $\frac{4 π}{18}$ or $\frac{2 π}{9}$ .

Ready for More?

1.

Consider a circle on a coordinate grid with its center at the origin $(0, 0)$ . Complete the following statement.

(Note: One ray of each of the angles will be the ray that extends from the center of the circle along the positive $x$ -axis, and the angles should be measured counterclockwise from this ray.)

The non-horizontal ray of angles measuring between $\underset{―}{}$ and $\underset{―}{}$ radians will lie in the first quadrant, between $\underset{―}{}$ and $\underset{―}{}$ radians in the second quadrant, between $\underset{―}{}$ and $\underset{―}{}$ radians in the third quadrant, and between $\underset{―}{}$ and $\underset{―}{}$ radians in the fourth quadrant.

2.

As you are responding to this prompt, consider this question: Is it possible for an angle to measure more than $2 π radians$ ?

Takeaways

Radians written in terms of $π$ allow

Radians written as decimals give

Visualizing angles measured in radians is assisted by knowing some benchmark angles, for example (list as many degree / radian equivalents as you can):

To convert from radians to degrees, I can

To convert from degrees to radians, I can

Lesson Summary

In this lesson, we learned how to approximate the size of an angle measured in radians, and we learned the radian measure for some familiar angles measured in degrees, such as 90° and 180°. We also learned how to convert between degree and radian measures.

Retrieval

1.

a.

If you purchased $π gallons$ of gasoline at the pump, how many gallons did you buy?

b.

If your employer said he would pay you $5 π dollars per hour$ , how much would you be making per hour?

c.

If an arc measures $π$ feet, how many feet long is it?

d.

If an arc measures $π radians$ , and the radius of the circle is $9 inches$ , how many inches long is the arc?

2.

Draw a central angle on the circle.
What do you need to know to draw a central angle?
Draw an inscribed angle that subtends the same arc as the central angle you drew.
What is the relationship between the measure of a central angle and the corresponding inscribed angle?