Ch 3 Radian Measure and the Unit Circle
3.1 Radian Measure
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.
C  2r
360  2 radians
Convert 45° to radians
Convert -270° to radians
Convert 249.8° to radians
1 radian 
180

9
Convert
radians to degrees
4
Convert 
5
radians to degrees
6
Convert 4.25 radians to degrees
If no unit of angle measure is specified,
then the angle is understood to be
measured in radians.
2
Find tan
3
Find sin
3
2
Find cos 
4
3
3.2 Applications of Radian Measure
The length s of the arc intercepted on a
circle of radius r by a central angle of
measure θ radians is given by the product of
the radius and the radian measure of the
angle.
s  r (θ in radians)
3
Find the arc length intercepted by
radians
8
Find the arc length intercepted by 144°
The north latitude of Reno NV is
approximately 40° while the north latitude
of Los Angeles is approximately 34°. Reno
is approximately due north of L.A. Given
the radius of the earth is approximately
6400km, find the straightline distance.
A rope is wound around a drum with radius
0.8725 ft. How much rope is wound when
rotated through 39.72°?
Two gears are adjusted so the smaller gear
drives the larger one. If the smaller gear
rotates through 225°, how many degrees will
the larger gear rotate?
A sector of a circle is the portion of the
interior of a circle intercepted by a central
angle.
The area of a circle is found by A  r 2 .
The area A of a sector of a circle of radius r
and central angle θ is given by the following
formula.
 1 2
A  r 2
 r
2 2
A center pivot irrigation system provides
water to a sector shaped field rotating
through 15°. Find the area supplied.
3.3 The Unit Circle and Circular Functions
A unit circle has its center at the origin and a
radius of 1 unit.
The trigonometric functions of angle θ in
radians found by choosing a point (x, y) on
the unit circle can be rewritten as functions
of the arc length s, a real number.
For any real number s represented by a
directed arc on the unit circle,
y
tan s 
sin s  y
cos s  x
x
1
1
x
sec s 
csc s 
cot s 
x
y
y
x 2  y 2  cos 2 s  sin 2 s  1
Since  1  x  1,  1  cos s  1, and
since  1  y  1,  1  sin s  1
domains:
sin and cos -  ,  

tan and sec - 
s
s


2
n

1


( x  0)
2

cot and csc - s s  n  y  0
3
Find exact value of sin
2
3
Find exact value of cos
2
3
Find exact value of tan
2
7
7
Find exact value of cos
and sin
4
4
5
Find exact value of tan 
3
2
Find exact value of cos
(120°)
3
calculator exercise
cos 1.85 cos 0.5149 cot 1.3209 sec  2.9234 


Approximate the value of s   0,  when
 2
cos s = 0.9685
3 

Find the exact value of s    ,  when
 2 
tan s = 1
The angle of elevation θ of the sun in the
sky at any latitude L can be found by
sin   cos D cos L cos   sin D sin L
In Sacramento, CA, L is 38.5° (0.6720 rad).
Find the angle at 3pm, D  0.1425,
ω=0.7854.
OQ
PQ
cos  
 OQ sin  
 PQ
OP
OP
VR
OV
tan  
 VR sec  
 OV
OR
OR
OU
csc SUO  csc  
 OU
OS
US
cot SUO  cot  
 US
OS
Suppose angle TVU measures 60°. Find
lengths of OQ, PQ, VR, OV, OU, and US.
3.4 Linear and Angular Speed
Linear Speed
d  rt
Angular Speed
speed 
distance
s
or v 
time
t


t
The human wrist can rotate through 90° in
0.045 seconds. Angular speed is:
Suppose P is on a circle with radius 10cm,
and ray OP is rotating with angular speed

18 radian/sec.
Find angle generated by P in 6 sec.
Find distance traveled in 6 sec.
Find linear speed of P in cm/sec.
A belt runs a pulley of radius 6cm at 80 rpm.
Find the angular speed in radians/sec.
Find the linear speed of the belt in cm/sec.
A satellite traveling in a circular orbit 1600
km above the earth takes 2 hr to make an
orbit. Radius of earth is about 6400 km.
Approximate the linear speed in km/hr.
Approximate the dist traveled in 4.5 hr.