Radians and Coterminal Angles
TASK ONE
Materials per group:
•
Pen or Marker
•
Ruler
•
String, approximately 1 foot
•
Protractor
•
Chart Paper
Instructions: (Check off each instruction as they are completed)
Draw a point in the middle of the chart paper.
Have one student hold one end of the string on the point drawn and another student hold
the other end of the string to the marker. Keeping the string tight, draw a circle.
Mark a point anywhere on the circumference of the circle.
Using the ruler, draw a radius connecting the center point to the point on the
circumference.
Place the string on the circumference of the circle, placing one end of the string on the
point on the circle.
Mark another point on the circle at the other end of the string.
Draw the radius from this point to the center.
Continue mapping the length of the string around the circumference of the circle making
sure that, as you go, you mark
§ the length along the circumference and
§ the radius
Use the protractor to obtain the approximate angle measure, to the nearest degree, for
each central angle in your circle.
List all angles here:
Discuss within your Group: What do you notice about your angles?
Do you think that groups with a larger radius will have larger, smaller or the same central
angles? Why? What if they had a smaller radius?
Based on the work you did, consider only the angles that were
approximately the same.
What statement can you make about a, b and c and the angle?
TASK TWO
When a=b=c, the central angle is equal to 1 radian or approximately _____ o .
One __________ is the measure of the
central angle…
…subtended by an arc equal in
length to the _________ of the circle.
In English: Consider the image on the right. If your arc and radii are the same
length, the central angle will be 1 radian.
1. Approximately how many times did the length of the radius fit onto the circumference?
2. How many radian angles fit inside one circle?
3. Analyze the work so far and make a connection
between the number of radians in a full circle to
the circumference. The radius has a value of r.
Arc Length
r
Central Angle
1 rad
2πr (circumference)
____ rads
Conclusion: 360o =
____ rads
SPECIAL NOTE:
Fften you’ll see radians shortened to rads. Sometimes, they will leave out the
units for radian completely. That means if you see 2π then it really means 2π
radians.
360o = 2π 4. Consider the following circle that represents
the one you created. Shade in 2 radians.
5. If a circle has 360o or 2π radians, how many radians would it take to match up to 180o?
Approximate Value
Exact Value
6. Figure out the missing radians values that match up to the angles on the circle graph.
7. Consider the previous question. Other than 0o = 2π, what is the easiest relationship to
remember?
8. How could you use this relationship to determine any missing value?
37o = _____ radians
π/5 radians = __________ o
TASK THREE
1. Johnny measures an angle and states that it is 5200.
How is this possible?
How would we normally name this angle?
What would 520o look like in radians?
2. Select an angle that is greater than 360o. Convert it to radians.
Coterminal Angles:
Angles that have the same terminal
arm.
3. How many Coterminal angles are there for 45o? How do you know?
4. Johnny makes a wild statement. He says that 30o is Coterminal with -330o. He creates the
following table to demonstrate other Coterminal angles.
Value
Coterminal His teacher agrees with him. Can
o
you figure out how negative angles
60
-300 o
o
o
work? Find a pattern.
90
-270
180 o
-180 o
73 o
-287 o
What statement can you make about negative angles?
Do you think this translates to Radians as well? Why or why not?
5. It would be difficult to list all of the possible angles, wouldn’t it? Let’s say, we’re starting
with 30o and we want to list the coterminal angles. We could say
30o + 1 revolution of 360o = 390o
30o + 2 revolutions of 360o =750o
30o + 3 revolutions of 360o = 1110o
30o + 1 revolution of -360o = -330o (aka 30o - 1 revolution of 360o = -330o)
30o + 2 revolutions of -360o = -690o (aka 30o - 2 revolutions of 360o = -690o)
Using a variable to represent the number of revolutions, can you determine
an expression to find all the positive coterminal angles for 300?
Negative coterminal angles?
Can you combine these into one statement?
This is called the General Form.
6. Select any angle. List all possible coterminal angles if -360o ≤ θ < 360o.
RECAP
Converting Radian to Degree: !!
210° !
!!!
!
Converting Degree to Radian: −60° PRACTICE
11. State if the given angles are coterminal. 12. Find a coterminal angle between 0° and 360°. !!
13. Find a positive and a negative coterminal angle. 14. Which of the following angles is coterminal with the angle ! ? Why? (There can be more than one!) !
!!
!!
!!
A. ! B. − ! C. ! D. !