Math factor

246 Pages·1996·16.2 MB·English

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Lesson 1:
Radian Measurement
Student Workbook
Produced by for
Liberia
EDUCATION
This guide accompanies Radian Measurement from the Math Factor series.
Writer: Wendy Stanhope
Editor: Wilma Gonsalves
Producer: Katherine Rankin
Educators in public educational institutions in Alberta may verbally quote, or use in
exams, excerpts from this publication, except material that has been credited to another
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The order number and title is
BPN #3689-09
Trigonometric and Circular Functions: Radian Measurement
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© 1996 Alberta Education
ISBN 1-896762-06-9
PRINTED AND BOUND IN CANADA
UNIVERSITY LIBRARY
UNIVERSITY OF ALBERTA
MATH FACTOR
TRIGONOMETRIC AND
Lesson 1: RADIAN MEASUREMENT
LEARNER
EXPECTATIONS
By the end of
Lesson 1 y ou should
be able to: define the radian measurement of a c entral angle in a c ircle
use the formula G = j t o solve for 6, a o r r
convert angle measurements between degree and radian measure and vice versa
apply this knowledge to solving problems involving sector area and angular velocity y
This lesson has
four parts: Defining One Radian
Converting
EXPLANATION OF ICONS
Building Blocks:
Do you remember most of y our Mathematics 20? This course builds upon many of those
skills and concepts. The Building Blocks icon appears in the workbook to indicate review
examples and/or exercises. To refresh your memory, do this section in the workbook
before watching the video.
Video:
Whenever this icon appears in the workbook, you should watch the indicated segment of
the video program. Preceding this icon, there is often a p roblem titled “Challenge
yourself.” Try the problem before watching the video. As you work through a p rogram
segment, stop the video to
• challenge yourself to find the solution before it is g iven
• write down key ideas and examples
• reflea on what is being taught
Challenge Yourself:
This icon sometimes appears on the television screen when a n ew example is starting.
When it appears, it i s suggested that you stop the video and try the example yourself
before proceeding. For your convenience, sometimes the example is already written in
the workbook. The solution will be given in the video.
Student Workbook:
This icon appears on the television screen at the end of teaching segments, when a new
skill or sub-skill has been taught The icon indicates that the workbook contains related
examples and/or exercises. Stop the video to study the examples and complete the
exercises. The video icon in the workbook will indicate when you should return to the
video and move on to learning a n ew skill.
Excellence:
This icon appears in the workbook to indicate advanced level examples or exercises that
you might find challenging. Test your new skills!
Textbook Reference:
This icon appears in the workbook to indicate additional examples and exercises that
can be found in the following three textbooks:
• M athematics 12 (Addison- Wesley)
• M athematics: Principles and Process 12 (Nelson Canada)
• H oltmath 12 (Holt, Rinehart and Winston oj Canada, Limited)
0
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Additional Examples
Mathematics 12: pp. 198, 199
Mathematics: Principles and Process 12: p. 181
Holtmath 12: p. 287
Exercises
1 . Cut off a p iece of string that is equal in length to the radius
of this circle. Use this string to draw an angle of one radian.
2. Using the string as a m easuring tool, how many lengths of
string are there around the circle's circumference?
Approximately how many radians are there
in 360°?
3. Complete the following definition.
An angle of one radian is defined as
following symbols represents in the formula.
a. 9
b. a
c. r
of 5 r ad?
4
.
In general , radians can be defined by the formula 9- j. Explain what each of the
5
.
If t he radius of a c ircle is 7 c m, then what is the measur e of the arc length for an angle
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Radian Measurement
the radian measure of this central angle?
6. The central angle of a c ircle of radius 10 cm subtends an arc length of 45 cm. What is
radian measure of this central angle?
central angle that subtends an arc length of 2kt1
7. The cen tral angle of a c ircle of radius r c m subtends an arc length of 2k r. What is the
2k rad?
^ EX ERCISE la SOLUTIONS are at the back of this workbook.
8
.A c ircle of radius r h as a c ircumference of 2kt. What is the degree measure of the
9
.Compare y our answers in q uestions 7 a nd 8. What degree measure is equivalent to
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Example 1
Convert 60° to radian measure.
Solution
Method 1
180° = /rrad
To convert degrees to radians, multiply \
rad
1° =
180 the number of degrees by 755- .
r \
n rad
60° =60°
= — r ad v#
3
Method 2
180'
n _ n
# Make equivalent ratios of radians to
180° 60° degrees: —k = Hn uummbbeerr ooff d rladgirarnes 0_r_
180° n- 60° k
using you can compare degrees
to radians.
n
3
Exact Radian 1 8M0e°a sure
§ i s the exact radian equivalent for 60°.
Approximate Radian Measure
o To express radians in decimal form,
replace n b y 3.14... by using the k
. 3.14
3 k button on your calculator.
= 1 .05 rad
60"
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Radian Measurement
Example 2
Convert jn rad to degree measure.
Solution
/rrad = 1 80° To convert exact radians to degrees,
replace n rad by 180c
f*rad = f( l80‘)
Exampl=e 1 3 53°
Convert 3.66 rad to degree measure.
Solution
Method 1
To convert decimal equivalent
K
1 rad = 1 801 radians to degree measure, multiply/
3.66 rad = 3 .66 If 1 2 0* 1 . J the number of radians by , *
where n = 3 .14... .
Method 2 = 2 10°
smmss 1 ' -
Make equivalent ratios of degrees to
degrees — ► 180° n draedgiraeness
radians — ► n 3.66 ^ \ radians or radians to degrees.
‘ § ' • . '
180° (3.66) = n n
180 ° ( 3.6 6) = n [ Solve for n.
K
180 ° ( 3.6 6) = n
f Using the n b utton on a c alculator,
3.14 L k = 3 .14.... A
n = 2 10°
Additional Examples
Mathematics 12: p. 200 (Question 3)
Mathematics : P rinciples and Process 12: pp. 181, 182
Holtmath 12: p. 287
0
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Radian Measurement
Exercises
10. Complete the following table.
Exact Radian Approximate Radian
Degrees Measure Measure
7T
2
K 3.14
it
2k 0° 6.28
270°
6
5.76
5 i t
T
225°
1 1 . Certain angles are identified on each of the following circles. Convert the measure of
the identified angles from degree to exact radian measure.
a. quadrant angles
b. angles with a r eference angle of 45°
ra
TRIGONOMETRIC AND CIRCULAR FUNCTIONS
Radian Measurement
c. angles with a reference angle of 30°
d. angles with a r eference angle of 60°
^ EX ERCISE lb SOLUTIONS are at the back of t his workbook.
Additional Practice
Mathematics 12\ p. 200 (Question 1), p. 201 (Questions 2-4)
Mathematics: Principles and Process 12\ p. 182 (Questions 1, 2, 5), p. 183 (Question 9)
Holtniath 12: p. 287 (Questions 3, 4)