COTERMINAL ANCLES In Example 1, the angles 500° and 140° are coterminal
because their terminal sides coincide. An angle coterminal with a given angle
can be found by adding or subtracting multiples of 360°.
3 ^ Define General Angles
^ • ^ and Use Radian Measure
(jEXAMPLE 2
Find one positive angle and one negative angle that are coterminal with
(a) - 4 5 ° and (b) 395°.
You used acute angles measured in degrees.
You will use general angles that may be measured in radians.
So you can find the area of a curved playing field, as in Example 4.
/ocabulary
filial side
linal side
ftndard position
iferminal
Find coterminal angles
Solution
There are many such angles, depending on what multiple of 360° is added or
subtracted.
a. - 4 5 ° + 360° = 315°
- 4 5 ° - 360° = -405°
In Lesson 13.1, you worked only with acute angles. In this lesson, you will study
angles with measures that can be any real numbers.
b. 395° - 360° = 35°
395° - 2(360°) = -325°
For foar Notebook
KEY CONCEPT
Angles in Standard Position
In a coordinate plane, an angle can be formed
by fixing one ray, called the initial side, and
rotating the other ray, called the terminal side,
about the vertex.
ibthematkal
ifalysis: 1.0 Students
nderstand the notion
f angle and how to
leasure I t In both derees and radians. They
an convert between
egrees and radians.
180°
An angle is in standard position if its vertex is at
the origin and its initial side lies on the positive
x-axis.
s
270°
GUIDED PRACTICE j for Examples 1 and 2
Draw an angle with the given measure in standard position. Then find one
positive coterminal angle and one negative coterminal angle.
The measure of an angle is positive if the rotation of its terminal side is
counterclockwise, and negative if the rotation is clockwise. The terminal
side of an angle can make more than one complete rotation.
EXAMPLE 1
1. 65°
Draw angles in standard position
b. 500°
c. -50°
Solution
Because 240° is 60°
more than 180°, the
terminal side is 60°
counterclockwise past
the negative x-axis.
Because 500° is 140°
more than 360°, the
terminal side makes
one whole revolution
counterclockwise
plus 140° more.
KEY CONCEPT
IV J '
^500°
:
•1
Define General Angles and Use Radian Measure
For Your Notebook
Converting Between Degrees and Radians
1
• " f
13.2
4. 740°
Because the circumference of a circle is 27rr, there
are 27r radians in a full circle. Degree measure and
radian measure are therefore related by the equation
360° = 2TT radians, or 180° = IT radiant.
Because —50°
is negative, the
terminal side is 50°
clockwise from the
positive x-axis.
1•y
—-^140°
V
3. 300°
RADIAN MEASURE Angles can also be measured in
radians. To define a radian, consider a circle with
radius r centered at the origin as shown. One radian
is the measure of an angle in standard position
whose terminal side intercepts an arc of length r.
Draw an angle with the given measure in standard position,
a. 240°
2. 230°
859
360
Degrees to radians
Radians to degrees
Multiply degree measure
IT radians
b
y
180°
Multiply radian measure
180°
by
7r radians'
Chapter 13 Trigonometric Ratios and Functions
EXAMPLE 3
HNC
nit"radians" is
omitted. For
ice, the measure
radians may be
Convert between degrees and radians
f EXAMPLE 4
SOFTBALL A softball field forms a sector with the
dimensions shown. Find the length of the outfield
fence and the area of the field.
Convert (a) 125° to radians and (b) — ~ radians to degrees.
a. 125°
, „ c o / IT radians \
125
I 180° )
b. - £ = (-JLradiansV 18 °° )
12 \ 12
A 7T radians/
in simply as —^-.
Solution
-15°
= ^
radians
36
Solve a multi-step problem
STEP 1 Convert the measure of the central
angle to radians.
AVOID ERRORS
90° = 9 o ° ( ^ radians \ = ^
I 180° I 2
You must write the
measure of an angle
in radians when using
the formulas for the arc
length and area of a
sector.
The diagram shovys equivalent degree
and radian measures for special angles
from 0° to 360° (0 radians to 1-n radians).
STEP 2
di
Find the arc length and the area of
the sector.
Arc length: s = rd= 180(-|) = 90TT «- 283 feet
Area: ,4 = ^ 0 = |(180) 2 /-|) = 8100-n- - 25,400 ft2
You may find it helpful to memorize the
equivalent degree and radian measures of
special angles in the first quadrant and for
• The length of the outfield fence is about 283 feet. The area of the field is about
25,400 square feet.
90° = ~ radians. All other special angles
• ^ are just multiples of these angles.
I
GUIDED PRACTICE
for Example 4
9. WHAT IF? In Example 4, estimate the length of the outfield fence and the
area of the field if the outfield fence is 220 feet from home plate.
GUIDED PRACTICE ] for Example 3
13.2 EXERCISES
Convert the degree measure to radians or the radian measure to degrees.
5. 135°
577
6. - 5 0 °
8.
HOMEWORK j Q _ WORKED-OUT SOLUTIONS
KEY;
on p. WS22 for Exs. 11,23, and 51
! • = STANDARDIZED TEST PRACTICE
EXS. 2,14,31,50, and 53
10
SKILL PRACTICE
SECTORS OF CIRCLES A sector is a region of a circle that is bounded by two radii
and an arc of the circle. The central angle 6 of a sector is the angle formed by the
two radii. There are simple formulas for the arc length and area of a sector when
the central angle is measured in radians.
1. VOCABULARY Copy and complete: An angle is in standard position if its
vertex is at the ? and its ? lies on the positive jc-axis.
2. * WRITING How does the sign of an angle's measure determine its direction
of rotation?
For Your Notebook
KEY CONCEPT
EXAMPLES
1 and 3
Arc Length and Area of a Sector
on pp. 859-861
for Exs. 3-14
The arc length s and area A of a sector with
radius r a n d central angle 6 (measured in
radians) are as follows.
VISUAL THINKING Match the angle measure wijh the angle.
3. -240°
A.
4. 600°
|y
B.
ength
Arc length: s = rd
Area: A = -1*6
13.2
Define General Angles and Use Radian Measure
861
862
Chapter 13 Trigonometric Ratios and Functions
5.
V
*y
C.
DRAWING ANGLES Draw an angle with the given measure in standard position.
6. 110°
10.
6TT
7. -10°
8. 450°
©S
12. -
9. -900°
5TT
13.
Eva,uate
13
3
Trigonometric
9
" * -* Functions of Any Angle
26TT
14. * MULTIPLE CHOICE Which angle measure is shown in
the diagram?
(A) -150°
<D 210°
< g ) 570°
( g ) 930°
You evaluated trigonometric functions of an acute angle.
You will evaluate trigonometric functions of any angle.
So you can calculate distances involving rotating objects, as in Ex. 37.
FINDING COTERMINAL ANGLES Find one positive angle and one negative angle
that are coterminal with the given angle.
15. 70°
16. 255'
19. f
20
7TT
6
17. -125°
18. 820°
21. 2 8 ^
22.
Key Vocabulary
• unit circle
• quadrantal angle
• reference angle
^
jndards)-
CONVERTING MEASURES Convert the degree measure to radians or the radian
measure to degrees.
23^40°
24. 315°
25. - 2 6 0 °
26. 500°
27. f
28. - • ?
29.
30. 14TT
15
Sir
Mathematical
Analysis: 9.0 Students
compute, by hand, the
values of the trigonometric functions and
the inverse trigonometric functions at various
standard points.
13ir,
31. * MULTIPLE CHOICE Which angle measure is equivalent to -=^=. radians?
(£> 30°
(g) 750°
( g ) 390°
Mathematical
Analysis: 2.0 Students
know the definition
of sine and cosine as
y- and x-coordinates of
points on the unit circle
and are familiar with the
graphs of the sine and
cosine functions.
Cg) 1110°
FINDING ARC LENGTH AND AREA Find the arc length and area of a sector with
the given radius r and central angle ft.
32. r = 4in., 0 = -£
33. r = 3 m , 0 = 4 ?
34. r = 15 cm, 0 = 45°
35. r = 12 ft, 0 = 150°
36. r = 18 m, 0 = 25°
37. r = 25 in., 0 = 270°
38. ERROR ANALYSIS Describe and correct
the error in finding the area of a sector
with a radius of 6 centimeters and a
central angle of 40°.
- 2 ,
*=
— i '( "=
—720
nnr, c—2
A
6 )>" (<4x'v>
0) =
m'
x:
40. sin
41. tan-£6
43. cot 4
44. cos
45. sin
3TT
Let 0 be an angle in standard position, and let (x, y)
be the point where the terminal side of 0 intersects
the circle x2 + y2 = r2. The six trigonometric
functions of 0 are defined as follows:
sin 0 =
esc 0 = —, y # 0
y
cos 0 = —
r
sec 0 = —, x + 0
x
tan 0 = i , x * 0
x
cot 0 = ^ , y ¥= 0
y
; jj These functions are sometimes called circular
functions.
Evaluate trigonometric functions given a point
Solution
Use the Pythagorean theorem to find the value of r.
r = V* 2 + /
4ir
15
= V(-4) 2 + 3 2 = V25 = 5
Using x = - 4 , y = 3, and r = 5, you can write
the following:
47. CHALLENGE A rotating object that passes through an angle 0 during time t
a
sin 0 = - = 1
has an angular velocity v given by the formula v = —. Find the angular
r
velocity of the hour hand, the minute hand, and the second hand on a
12 hour clock. Give all answers in degrees per hour.
13.2 Define General Angles and Use Radian Measure
r rout mistook,
General Definitions of Trigonometric Functions
Let ( - 4 , 3 ) be a point on the terminal side of an
angle 0 in standard position. Evaluate the six
trigonometric functions of 0.
42. sec —
46. esc
KEY CONCEPT
(iEXAMPLE 1
EVALUATING FUNCTIONS Evaluate the trigonometric function using a calculator
if necessary. If possible, give an exact answer.
39. cos-^-
You can generalize the right-triangle definitions of trigonometric functions from
Lesson 13.1 so that they apply to any angle in standard position.
5
cos 0 = - = - 4
esc e = — = —
y
3
863
866
Chapter 13 Trigonometric Ratios and Functions
tan 0 = i- = —i
cot 0 = i = —±