Download Trigonometry Section 3

Trigonometry Section 3.1
Convert each degree measure to radians.
1.
23.0143
2.
30
3.
39
4.
4230
5.
85.04
6.
120
7.
17450
8.
270
9.
315
10.
390
11.
450
12.
600
Convert each radian measure to degrees. Write answers to the nearest minute or thousandths of
a degree.
13.
 3.47189
14.
4
15
15.
0.3417
16.
7
10
17.
4
3
18.
1.74
19.
7
4
20.
11
6
21.
5
2
22.
8
3
23.
15
4
24.
5
Find the EXACT value of each of the following WITHOUT using a calculator.
25.
sin
2
3
26.
cos
5
6
27.
tan
29.
sec
3
4
30.
csc
7
6
31.
  4 
cos

 3 
33.
  11 
csc

 4 
34.
  
tan 

 6 
35.
csc
37.
cos 3
38.
  23 
cot 

 6 
39.
sin

4

2
13
4
5
3
28.
cot
32.
sin 
36.
sec
40.
  13 
sec

 3 
11
6
Trigonometry Section 3.2
Find the length of the arc intercepted by a central angle
2
radians
3
1.
r  12.3cm, 
3.
r  253m, 
5.
r  4.82m,  60
2
radians
5
 in a circle of radius r .
11
radians
10
2.
r  0.892cm, 
4.
r  120mm, 
6.
r  71.9cm,  135

9
radians
Find the measure (in radians) of a central angle that intercepts an arc of given length in a circle
with given radius.
7.
s  5in, r  2in
s  6m, r  4m
8.
9.
s  30cm, r  5cm
Find the radius of a circle in which a central angle (in radians) is given in radians and intercepts
an arc of given length.
10.
  2radians , s  3 ft
12.
11.


5

3
radians , s  6cm
4
radians , s  4in
Find the distance in kilometers between each pair of cities, assuming they lie on the same northsouth line.
13.
Panama City, Panama 9 N
Pittsburgh, Pennsylvania 40N
15.
Assuming that Earth is a sphere of radius 4000 miles, what is the difference in the
latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is about 450
miles due north of Annapolis?
14.
New York City, New York
Lima, Peru 12S
41N
Work the applied problems.
16.
a)
How many inches will the weight in the figure rise if the pulley is rotated through
an angle of 7150 ?
b)
Through what angle, to the nearest minute, must the pulley be rotated to raise
the weight 6in ?
9.27 in
17.
The rotation of the smaller wheel in the figure causes the larger wheel to rotate. Through
how many degrees will the larger wheel rotate if the smaller one rotates through 60.0 ?
5.23cm
8.16cm
Find the area of a sector of a circle having radius r and central angle
5
radians
6
18.
r  29.2m, 
20.
r  52cm, 
22.
r  12.7cm,  81
3
radians
10
.
2
radians
3
19.
r  59.8km, 
21.
r  25mm, 
23.
r  18.3m,  125

15
radians
Find the measure (in radians) of a central angle of a sector of given area in a circle with given
radius.
24.
A  16in 2 , r  3.0in
25.
A  25in 2 , r  10in
Find the radius of a circle in which a central angle is given in radians and determines a sector of
given area.
26.


4
radians , A  36 ft 2
27.


6
radians , A  64m 2
Work the applied problem.
28.
The figure shows Medicine Wheel, a Native American structure in northern Wyoming.
This circular structure is perhaps 200 years old. There are 32 spokes in the wheel, all
equally spaced.
a)
Find the measure of each central angle
in degrees and radians.
b)
If the radius of the wheel is 76 ft , find
the circumference.
c)
Find the length of each arc intercepted
by consecutive pairs of spokes.
d)
Find the area of each sector formed by
consecutive spokes.
29.
A sprinkler on a golf green is set to spray water over a distance of 15 meters and to
rotate through an angle of 140 . Find the area of the region.
Trigonometry Section 3.3
Find the EXACT circular function value for each of the following WITHOUT using a calculator.
1.
  5 
sin 

 6 
2.
cos
3
4
3.
tan
4
3
4.
  7 
cot

 4 
5.
sec 2
6.
csc
17
3
7.
  5 
cos

 2 
8.
sec
9.
tan
5
2
10.
  5 
sin 

 3 
11.
cot   
12.
  7 
csc

 6 
13.
cos
7
6
14.
  3 
sec

 2 
15.
csc 3 
16.
sin
17.
cot
11
6
18.
tan 2
19.
sec
13
4
20.
  9 
cos

 4 
8
3
3
2
Use a calculator to find an approximation for each circular function value.
21.
sin 7.5835
22.
cot 0.6632
23.
cos 3.8426
24.
tan 0.9047
25.
csc1.3875
26.
sec 6.6701
27.
sin  2.2864
28.
tan 6.4752
29.
cot 7.4526
Find the value of
 
s in the interval 0,  that makes each statement true.
 2
30.
sin s  0.99184065
31.
cot s  0.29949853
32.
cos s  0.78269876
33.
csc s  1.0219553
34.
tan s  0.21264138
35.
sec s  1.0219553
Find the EXACT value of
NOT use a calculator.
36.
sin s 
s in the given interval that has the given circular function value. Do
1  
,
,
2  2 
37.
cos s 
For each value of s , use a calculator to find
which quadrant an angle of s radians lies.
39.
s  49
1
,
2
 3 
 , 2  38.
 3

tan s  1,  ,2 
 2

sin s and cos s , then use the results to decide in
40.
s  65
Trigonometry Section 3.4
Use the formulas to find the value of variables.

1.
Find  if  
2.
Find  if  
3.
Find t if  
4.
Find  if   0.90674radians / min, t  11.876 min
5.
Find v if r  12m,  
6.
Find  if v  9m / sec, r  5m
7.
Find  if v  107.692m / sec, r  58.7413m
8.
Find s if r  6cm,  
9.
Find t if s 
10.
Find  if s 
4
radians / sec, t  5 min
2
radians , t  10sec
5
3

radians ,  
radians / min
8
24
2
radians / sec
3

3
radians / sec, t  9 sec
12
3
2
m, r  m,  
radians / sec
5
2
5
3
km, r  2km, t  4 sec
4
Work the following problems.
11.
Find
 for the minute hand of a clock.
12.
Find
v for the tip of the minute hand of a clock, if the hand is 7cm long.
13.
Find
 for a line from the center to the edge of a phonograph record revolving 33
times per minute.
14.
Find
v for a point on the tread of a tire 18cm , rotating 35 times per minute.
1
3
15.
The Earth travels about the Sun in an orbit that is almost circular. Assume that the orbit
is a circle, with a radius 93,000,000miles .
a)
Assume that a year is
movement in one day.
16.
365 days, and find  , the angle formed by the Earth’s
b)
Give the angular velocity in radians per hour.
c)
Find the linear velocity of the Earth in miles per hour.
A circular power saw has a 7
1
inch diameter blade that rotates at 5000 revolutions per
4
minute.
a) Find the angular velocity of the saw blade in radians per minute.
b) Find the linear velocity (in feet per minute) of one of the 24 cutting teeth as they
contact the wood being cut.
17.
A car is moving at a rate of 65 miles per hour and the diameter of its wheels is 2 feet.
a) Find the number of revolutions per minute the wheels are rotating.
b) Find the angular velocity of the wheels in radians per minute.
TRIGONOMETRY
PRACTICE TEST:
Chapter 3
NAME: ____________________
Radian Measure and the Circular Functions
In order to receive full credit, you must show your work!!
Convert each of the following degree measures to radians. Leave answers as multiples of
1.
45
2.
210
3.
1020
6.
21
5
.
Convert each of the following radian measures to degrees.
4.
11
6
5.
8
3
Find the EXACT value of each of the following WITHOUT using a calculator.
7.
cos
5
4
8.
sin

3
9.
tan
5
6
10.
csc
3
2
11.
cot
4
3
12.
sec
2
3
13.
tan
7
4
14.
sin
11
6
15.
cos 
Use a calculator to find an approximation for the circular function values. Be sure your calculator
is set in radian mode.
16.
sin 1.0472
17.
sec 0.4864
18.
cot 3.8426
 
that makes each of the following true.
 2 
Find the value of s in the interval 0,
19.
cos s  0.92500448
20.
csc s  1.2361343
21.
tan s  4.0112357
Find the EXACT value of s in the given interval that has the given circular function value. Do
NOT use a calculator.
22.
cos s 
 2  
, ,
2  2 
24.
23.
sin s 
 3  3

,  ,2 
2 2

tan s 
3  3 
, ,
3  2 
Solve each problem.
25.
A circle has a radius of 8.973cm. Find the length of an arc on this circle intercepted by a
central angle of 49.06 .
26.
A central angle of
7
radians forms a sector of a circle. Find the area of the sector if
4
the radius of the circle is 28.69in.
27.
28.


Find the measure of the central angle  radians AND the area of the sector for a circle
with a radius of 2m and an arc length of 1.5m.
Find the distance in kilometers between Farmersville, California, 36 N and Penticton,
British Columbia, 49N , assuming they lie on the same north-south line. (Assume that
the radius of the Earth is 6400km)
29.
Find s if r  11.46cm,   4.283radians / sec, and
30.
Find
31.
Find r if s 
32.
Find v for the tip of an airplane propeller 3m long, rotating 500 times per min.

t  5.813sec .
if v  18 ft / sec and r  3 ft .
216
2
yd ,  
radians / sec and t  12 sec .
5
5
Solutions
1.

4
2.
7
6
3.
17
3
5.
480
6.
756
7.
cos
8.
sin
11.
cot
14.
sin
17.
sec0.4864  1.131194347
18.
cot 3.8426  1.184817486
19.
s  0.3897489421
20.
s  0.9424039526
21.
s  1.326476844
24.
s
27.
  0.75rads, A  1.5m2
28.
s  1452  1500km
29.
s  285.32  285.3cm
30.
 6
31.
r  9 yds
32.
v  4712  4700
4.
330
5
2

4
2
9.
tan
5
3

6
3
10.
csc
3
 1
2
4
3

3
3
12.
sec
2
 2
3
13.
tan
7
 1
4
11
1

6
2
15.
cos   1

3

5
3
3
2
25.
22.
s
s  7.683cm
16.
sin1.0472  0.8660266282
3
4
23.
26.
s
7
6
A  2262.22  2263in 2
rads
sec
m
min