In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name:

So far, we have measured angles in degrees. Because of the arbitrary nature of degree measurement, another measurement can be used called radians.

A radian is a unit of angle measure based on arc length.

An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a

measure of 1 radian.

If a central angle θ in a circle of radius r, cuts off an arc of length s, then the measure of θ in radians is:

θ= sr

In Class Examples:

1. Los Angeles & New York City are approximately 2,500 miles apart on the surface of the earth. Assuming that the radius of the earth is 4,000 miles, find the radian measure of the central angle with its vertex at the center of the earth that has LA on one side and NYC on the other.

One full rotation of a circle is equivalent to the circumference of the circle. This allows us to convert a 360 degree rotation to radians:

θ= sr

2. Find the radian measurement of the given angles: a) 30o b) 45o c) 60o d) 90o

e) 180o

This conversion can be generalized by a simple formula:

To convert degrees to radians, multiply by__________________

To convert radians to degrees, multiply by__________________

In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name:

In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name:

UNIT CIRCLE PAPER PLATE ACTIVITY: Part 2 (Link to video: http://bit.ly/1O7IuRp)

1. RADIAN: The measure of an angle in which the length of the radius equals the length of the arc cut by the angle. Write it on the back of your paper plate.

2. Take your piece of string and measure the length of the radius. Put your edge of the string on the origin and then stretch it along the initial ray. Mark the length on the string with a dark pen where it touches the point (1, 0).

3. Now with one end of the string on the point (1, 0) and the string wrapped along the unit circle, mark the point on the circle where the arc length equals the radius using your string.

4. Mark equal lengths around your unit circle as shown.

5. That angle that you created is one radian! Label the an-gle 1 radian on your unit circle.

6. How many radius lengths are there in the circumference of a circle?

7. Fold your paper plate in half and look at just QI and QII. How many radius lengths in one half the circumference of the circle?

8. Now we have three labels for each point. A coordinate la-bel (1, 0), a degree label 0 °∨360 °and a radian label 0∨2π radians.

9. Label each point in terms of radian measure.

In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name:

A circle centered at the origin can be represented by the relation x2 + y2=r 2 where r is the radius of the circle. Draw each circle:

1. x2 + y2=16 2. x2 + y2=25

Using the equation of the unit circle, x2 + y2=1, find the unknown coordinate of each point. Is there more than one unique answer?

3. ( 12, y ) 4. (x , √3

2 ) 5. . (x , √22 )

Convert each to degrees. Convert each to radians.

6. 11π12 7.

−π3 8. 190° 9. -315°

In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name:

Give a positive and a negative coterminal angle for each.

10. 10° 11. 3π5

12. The minute hand of a clock travels how many radians in 10 minutes? In 7 min-utes?

13. The minute hand on Big Ben’s Clock Tower in London is 14 ft long. To the nearest tenth of a foot, how far does the tip of the minute hand travel in 1 minute?

14. A tire of a car makes 653 complete rotations in 1 min. The diameter of the tire is 0.65 m. To the nearest meter, how far does the car travel in 1 s?

In Class 3 Community High SchoolAlgebra 2: Unit 8, The Unit CircleStudent Name: