Trigonometry
Unit 2 – Applications
Homework Problems
1. Find the exact length of the arc
intercepted by the central angle.
Name
3. Find the measure of each central angle in
radians, and to the nearest tenth of a
degree.
2. Find the radius of the circle.
4. Find the area of the sector.
5. Find the length of the arc intercepted by a
2
central angle  
radians in a circle of
3
radius 12.3 cm.
7. Find the area of a sector of a circle having
5
radius 29.2 m and central angle  
.
6
9.
6. Find the length of the arc intercepted by a
central angle   60 in a circle of radius
4.82 m.
8. Find the measure (in radians) of the
central angle that makes a sector of area
3 units2 in a circle of radius 2 units.
Find the distance in kilometers between New York City, New York and Lima, Peru, if NYC is
located at latitude 41o N and Lima is at latitude 12o S, assuming they lie approximately on
the same north-south line. Use 6400 km as the radius of the earth.
10. Madison, South Dakota and Dallas, Texas are 1200 km apart and lie on the same northsouth line. The latitude of Dallas is 33o N. What is the latitude of Madison? Use 6400 km as
the radius of the earth.
11. How many inches will a weight rise if the pulley is rotated through an
angle of 71o50’?
Through what angle, to the nearest minute, must the pulley be rotated
to raise the weight 6 in?
12. Find the radius of the larger wheel in the figure if the smaller wheel rotates 80.0o when the
larger wheel rotates 50.0o.
13. The Ford Model A, built in 1928 to 1931, had a single windshield wiper on the driver’s side.
The total arm and blade was 10 in long and rotated back and forth through and angle of 95o.
The shaded region in the figure is the portion of the windshield cleaned by a 7-in. wiper
blade. What is the area of the region cleaned?
14. A frequent problem in surveying city lots and rural lands adjacent to curves of highways
and railways is that of finding the area when one or more boundary lines is the arc of a
circle. Find the area of the lot shown in the figure. (Source: Anderson, J. and E. Michael,
Introduction to Surveying, Mc-Graw-Hill, 1985)
15. Nautical miles are used by ships and airplanes. They are different from statute miles, which
equal 5280 ft. A nautical mile is defined to be the arc length along the equator intercepted
by a central angle AOB of 1 minute, as illustrated in the picture. If the equatorial radius of
Earth is 3963 miles, use the arc length formula to approximate the number of statute miles
in 1 nautical mile. Round your answer to two decimal places.
16. The speedometer of Terry’s small pick-up truck is designed to be accurate with tires of
radius 14 in. Find the number of rotations of a tire in 1 hour if the truck is driven at 55 mph.
Suppose that oversize tires of radius 16 in are placed on the truck. If the truck is now driven
for 1 hour with the speedometer reading 55 mph, how far has the truck gone? If the speed
limit is 55 mph, does Terry deserve a speeding ticket?
Angular and Linear Speed
17. What is angular speed of the hour hand of a clock in radians per hour?
18. What is speed (in mm/sec) of the tip of the second hand of a clock, if the hand is 28 mm
long?
19. A 90-horsepower outboard motor at full throttle will rotate its propeller at 5000 revolutions per
min. What is the angular speed of the propeller in radians per second?
20. The tires of a bicycle have a radius of 13 inches and are turning at a rate of 200 revolutions
per minute. How fast is the bicycle moving in miles per hour?
21. Mars rotates on its axis at a rate of about .2552 radian per hour. Approximately how many
hours are in a Martian “day” (1 rotation)?
22. Earth travels about the sun in an orbit that is almost circular. Assume that the orbit is a circle
with radius 93,000,000 mi. It’s angular and linear speeds are used in designing solar-power
facilities.
a) Assuming that year is 365 days, find the angle in degrees formed by Earth’s movement
in one day to three decimal places.
b) Give the angular speed in degrees per week to three decimal places.
c) Find linear speed of Earth in miles per hour.
23. Earth revolves on its axis once every 24 hours. Assuming that Earth’s radius is 6400 km, find
the following.
a) angular speed of Earth in radians per hour and degrees per hour.
b) linear speed at the North Pole and South Pole in km/hour
c) linear Speed at Quito, Ecuador, a city on the equator in km/hour
d) BONUS: linear speed at Salem, Oregon (halfway from the equator to the North Pole)
24. Two pulleys have radii 15 cm and 8 cm, respectively. The larger pulley rotates 25 times in 36
sec. Find the angular speed of each pulley in radians per second and degrees per second.
25. A thread is being pulled off a spool at a rate of 59.4 cm per sec. Find the radius if the spool if
it makes 152 revolutions per min.
26. A railroad track is laid along the arc of a circle of radius 1800 feet. The circular part of the
track subtends a central angle of 40 . How long (in seconds) will it take a point on the train
traveling 30 mph to go around this portion of the track?
27. The shoulder joint can rotate at about 25 radians per sec. If a golfer’s arm is straight and the
distance from the shoulder to the club head is 5 ft, estimate the linear speed of the club
head from shoulder rotation in miles per hour. Round to the nearest whole number.
(Souce: Cooper, J. and R. Glassow, Kinesiology, Second Ediition, C.V. Mosby, 1968).
ANSWERS:
1) 2 units 2) 8 units
3) 1.5 radians, 85.9
4) 6 units2 5) 25.8 cm 6) 5.05 m
7) 1116.1 m2
8)1.5 radians
9) 5900 km 10) 44 N
11) a. 11.6 in b. 37 05 '
12) 18.7 cm 13) 75.4 in2 14) 1900 yd2 15) 1.15 mi 16) a. 39, 616 rotations b. 62.9 mi

17)
radian per hour
18) 2.9 mm/sec
19) 523.6 radians/sec
20)15.5 mph
6
21) 24.62 hour
22) a. .986 b. 6.904 per week c. 66,700 mph

23) a.
radian/hr
b. 0 km/hr
c. 1676 km/hr
d. 1200 km/hr
12
24) Larger Pulley: 4.36 radians/sec; 250 /sec Smaller Pulley: 8.18 radians/sec; 469 /sec
25) 3.73 cm 26) 29 sec 27)85 mph