Name:
Investigating Radian Measure
Connection to Astronomy: Astronomers do not mark the position of stars by noting the position of a star at one time, noting its position some time later, and then drawing a straight line between those two positions to track the star. Instead, between those two positions, they draw an arc whose length can be measured in the same units as the distances to the stars from the observer. Over time instead of thinking of the arc being defined by the angle, they came to think of the angle being defined by the arc and that led to radian measure.
arc length
r
r
Observer
Investigating Radian Measure
1. Use a compass to construct a circle with radius 5cm.
2. Draw the radius. Label the center C and the other end A.
3. Using a piece of string, measure/record the circumference of your circle. C =
4. Measure/record the radius of your circle. r =
5. Cut a piece of string the length of the radius.
6. Starting from point A, lay out the radius-length string along the
circumference of the circle. Mark B on the circle at the end of the string.
7. Draw a line from the center of the circle to the point on the circle you marked.
What is this line segment CB called?
How does the length of CB compare to the length of CA?
How do the lengths of CB and CA compare with the arc length, AB?
8. The central angle formed by the two radii CA and CB has an
angle measure 1 radian. Label this on your circle. In your own words,
what is the definition of an angle measuring 1 radian? Write it in the box below:
9. Use a protractor to measure the angle in degrees. Record the value. 1 radian = °
10. Compare the measure of the angle you drew with the measure of the angle
drawn by a classmate with a radius of a different size. Is the angle smaller? bigger? the same size?
11. What can you conclude about an angle of 1 radian?
12. Using the radius-length string, continue to wrap the string around the
edge of the circle, marking its ending locations. Draw in the radii.
About how many radii would it take to get around the whole circle?
13. Likewise, approximately how many radians are in a complete circle?
Approximately, how many radians are in a semi-circle?
Can you think of a mathematical value or constant that is close to
this latter number?
14. What is the formula for the circumference of a circle, in terms of
its radius r? C =
15. Since you only estimated the number of times the string could be wrapped around the circle, use your more precise circumference and radius measurements to calculate the number of radians.
16. Using the formula above, replace radius with r, and circumference with to calculate exactly how many radians there must be in a complete circle? ( Cross out any common values in the numerator and denominator)
17. How many degrees are in a complete circle?
18. If the number of radians in a complete circle is 2π,
how many radians are in a half circle?
19. How many degrees are in a half circle?
20. Use the information from the 3 previous steps to write two equivalence
relationships between radians and degrees.
21. Can you think of a method to convert back and forth between radians and degrees?
22. Convert each from degrees to radians. SHOW YOUR WORK.
- 60°
- 135°
- 206°
23. Convert each from radians to degrees. SHOW YOUR WORK.
-
- 1.62
24. Though the above gives us a way to convert between radians and degrees algebraically, we want to be able to both do these conversions in our heads as well as be able to quickly draw a sketch of an angle in radian measure the same way we can draw a sketch of an angle measured in degrees. We want to learn to think in radians. If you wanted to draw a sketch of a 60° angle your thinking might be: “How does 60° compare to 180°? 60° is one-third of 180°, so I need to draw and angle that’s one-third of a straight angle.” Label each circle in degrees and radians.
25. Before you are asked to sketch angles of various measures and to convert them to the other unit (Radians Degrees), several conventions need to be stated.
a. Angles are measured as the rotation from an initial ray to a terminal ray.
b. An angle is said to have positive measure if the rotation is counterclockwise.
c. An angle is said to have negative measure if the rotation is clockwise.
26. Draw a sketch of each of the following angles. Remember a straight angle has measure 180° or π radians. Use these facts as the reference point for drawing these angles. If the angle is given in degrees, find its equivalent radian measure. If the angle is given in radians, find its equivalent degree measure. For each conversion include the factor by which you are multiplying and give exact answers. Show your work for the conversions.
1
- –90°
- 315°
1
27. In determining the number of radians in a complete circle is 2π, we implicitly used the relationship that # radians in an angle = # radii in the circumference = arc length around circle
length of the radius
Recall from geometry that an angle with vertex at the center of a circle is called a
central angle. It makes sense for any central angle of any arc length: [*]
# radians in an angle = arc length .
length of the radius
In symbols, if the # radians in an angle = measure of the angle in radians = θ*,
then for any central angle with radius r and arc length s, θ = s / r. Use algebra
to solve this equation for s. Record your answer.
28. We have just determined and important relationship. Reiterate it below:
This relationship has many practical applications and is helpful in a variety of
problem-solving situations.
29. Returning to the relationship written in the form θ = ______we can see that if s and r are measured in the same units, the radian is unit-neutral, the length units cancel. For example if, r = 6 cm and θ = 4 radians, then s = 24 cm (not 24 radian-cm).
30. For homework read over this packet and rewrite formulas and conversions in your notebook.
1
[*] θ is one of several Greek letters used to denote angles. It is pronounced “theta”.