Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

5 – Trigonometric Functions Problem Set 5-1 1. Find length of the arc of an 870 central angle in a circle with radius 2 m.

2. Since its restoration in 1990, The Tower of Pisa

tilts at an angle of about 4o. The height of the tower in the center is about 56.28 m. How far is the top of the tower (in the center) from where it would be if the tower were perpendicular to the ground? Source: Quakr Viewer, Tower of Pisa, http://www.quakr.co.uk/quakr/xtech07/images_katie/16_leaningtower_angles.jpg, 12/19/11

3. In modern cataract surgery (Phacoemulsification), the ophthalmologist makes an incision that is 6% of the circumference of the cornea. If the diameter of a patient’s cornea is 8 mm, what is the length of the incision?

4. The meter was originally intended to be 1/10,000,000 of the distance from the North Pole to the equator. In 1799 two Frenchmen, Jean-Baptiste-Joseph Delambre and Pierre-François-André Méchain, tried to determine the distance from the North Pole to the equator along one meridian (keeping the longitude constant) by measuring a portion of the distance (Dunkirk, France to Barcelona, Spain) and used the idea of part is to whole (angle) as part is to whole (arc) to find an estimate of the distance from the North Pole to the Equator. Watch the video, “10,000,000 m with Google Maps”. Pick two places other than the ones on the video, and complete the proportion

090

angle between the cities distance between the cities

x

Solve for x and it should be approximately 10,000,000 m.

5. Find the area of the sector whose central angle is 1330 in a circle whose radius is 11 mm. Draw a circle and label the angle, the sector, and the radius.

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

6. The image is a satellite picture of Neligh, Nebraska (lat/long= 42.266766,-97.984922); the circle is evidence of “center pivot irrigation”. (To get a sense of scale, about 70 football fields would fit in this circle.) The radius of the circle is about 368 meters and the central angle of the part that was irrigated (the green portion) is about 315o. What is the area of the irrigated portion of the irrigated sector?

7. Calculate the number of square inches per dollar for each pizza. Which restaurant gives you a better deal?

Pizza Hut Large are 16” diameter. Source: Pizza Hut, http://www.pizzahut.com/deals.html, 12/20/2011 Domino’s Large are 14” diameter. Source: http://www.dominos.com/, 12/20/2011

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

8. According to the SIPRI database, the US spends 43% of the global expenditures on military. What central angle for the US needs to be made in order to accurately draw the pie chart?

9. You are involved in community service, tutoring students in a junior high after school program. Your tutee asks, “what is π?”. Give a succinct answer.

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-2 1. What is the length of the side labeled x?

2. The picture below is a Google Earth image of the famous Flatiron building in Manhattan. The roof of the building is a right triangle with the shortest side (on 22nd Street) 87 ft and the second shortest side (on 5th Avenue) 173 ft. a. What is the length of the largest side of the building? b. What is the measure of the smallest angle?

Source: Answers/Yahoo.com, Flatiron Building,

http://answers.yahoo.com/question/index?qid=20090105192319AAiw8li, 12/19/11

3. In ΔABC: a. What is the length of AC?

b. What is the length of BC?

4. Road signs warn of large hills and do so by indicating the grade. Grade is simply a ratio of the vertical distance to the horizontal distance. (The car travels the hypotenuse.) If a sign shows a 3% grade down hill, what is the angle of depression?

1

θ

B

C A

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

5. (If someone is 5 ft 9 in tall, the person’s eyes would be about 5 ft 5 in above the ground.) Someone whose eyes are 5 ft 5 above the ground is standing on the edge of the calm ocean and looks out to sea. (For this problem and the next, assume that the earth is a sphere.) a. How far away is the horizon?

b. How far away is the horizon if you stand on your tip toes (makes you 3

inches taller)?

c. How far away is the horizon if your eyes are x inches above the ground?

6. Imagine that a string goes all the way around the earth on the equator. a. How long is the string?

b. If the earth expanded so that its radius was 1 meter larger, how much

longer would the string need to be so that it would go around the earth on the equator?

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-3

1. Given ABC below. Find the length of BC.

2. In a water molecule (H2O) the angle formed by

H-O-H is about 104.45o and the O-H length is

about 0.9584 o

A . What is the H-H length?

(note: 10-8 cm = 1o

A or 1 angstrom)

Source: Wikimedia Commons, http://commons.wikimedia.org/wiki/File:Water_molecule_dimensions.png, 12/20/11

A

B

C

15

21o

23

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

3. The North Carolina Research Triangle is formed by UNC, Duke, and NC State. What is the angle of the triangle with Duke as the vertex?

4. In triangle ABC, a=24, c=32 and mB = 1150. Find the length of side b.

5. In ΔBFM, mB = 400, b=6, m=8. Use the Law of Cosines and the quadratic formula to find all possible values for f.

6. In ΔBFM, mB = 400, b=3, m=8. Use the Law of Cosines and the quadratic formula to find all possible values for f.

7. In ΔBFM, mB = 400, b=8, m=8. Use the Law of Cosines and the quadratic formula to find all possible values for f.

8. In ΔBFM, mB = 300, b=8, m=16. Use the Law of Cosines and the quadratic formula to find all possible values for f.

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-4 1. Carefully draw a graph of y = cos(x) on the "restricted domain".

2. Carefully draw a graph of y = cos-1(x).

3. What is the domain and range of the inverse Cosine Function?

4. Given f(x) = cos-1(x), find exact value, giving your answer in radians.

a. f 1 b. 2

f2

c. f 0

5. Given f(x) = cos-1(x), find exact value, giving your answer in degrees.

a. 2

f2

b. f 1

6. The distance from Miami to Bermuda is 1,041 miles, the distance from

Bermuda to San Juan is 971 miles, and the distance from Miami to San Juan is 975 miles. What is the angle formed with Miami at the vertex?

Image source: Bermuda Styles, The Bermuda Triangle, http://www.bermudastyles.com/bermuda/images/the-bermuda-triangle.jpg, 12/1307

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-5

1. In ∆ABC, a = 15, A = 26o, B = 76o. Find b.

2. In ∆ABC, A = 40o and a = 14. Find B if b = 19.

3. In Washington DC a 132o is formed using the White House (WH) as the vertex of an angle, looking out to Washington Circle (WC) and Mount Vernon Square (MV). It is 2,100 m from WC to MV and 1,200 m from MV to WH. What is the angle formed with WC at the vertex, looking out at WH and MV?

Map Source: David Halstead, DC Sightseeing Map, http://sc94.ameslab.gov/TOUR/tour.html, 8/12/03

4. In ΔFST, mF=340, FS=16, ST=10. Find mT.

5. The Google Earth picture below is in El Giza Egypt with the Pyramid of Khafre on the left and the Great Sphinx on the right. When standing at the Great Sphinx, 510 m from the base of the Pyramid, the angle of elevation to the top of the Pyramid is 13.19o. When standing at the base of the Pyramid, the angle of elevation is 51.82o. What is the slant height of the Pyramid?

WC MV

WH

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-6 1. Carefully draw a graph of y = sin(x) on the "restricted domain".

2. Carefully draw a graph of y = sin-1(x).

3. What is the domain and range of the Inverse Sine Function?

4. Given f(x) = sin-1(x), find exact value, giving your answer in radians.

a. f 1 b. 2

f2

c. f 0

5. Given f(x) = sin-1(x), find exact value, giving your answer in degrees.

a. 2

f2

b. f 1

6. In ABC, mBAC = 250, AB = 9 cm, and BC = 5 cm.

Find all possible values for mACB.

7. In ABC, mABC = 290, AC = 6.9 cm, and BC = 10.2 cm. Find mBAC.

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-7 Mixed Review of Right Triangle Trig, LOS, and LOC Draw the triangle as best as you can to scale and answer the question as completely as possible. If there is no answer or more than one answer you need to say so. Note the approximate answers are given in jumbled order below. If you don’t get one of the answers as your approximation, then you probably made a mistake. 36.90 300 9.2 1260 7.10 no triangle 127.8 20.20 no triangle 32.50 and 147.50 106.50

1. In ΔEDF, mE is 270, f=7, e=7. Find mD.

2. In ΔPQR, r=54, mP = 900, mQ = 65. Find p.

3. In ΔABC, mB = 290, b=1, a=9. Find mA.

4. In ΔDBF, f=9, mD = 800, b=4. Find d.

5. In ΔNPA, p=28, mP=200, a=44. Find mA.

6. In ΔKYD, mK is 900, d=3, y=4. Find mD.

7. In ΔGTB, g=234, mT = 1120, t=220. Find mG.

8. In ΔTEK, e=9, mK = 600, t=18. Find mE.

9. In ΔRFH, r=7, f=9, h=4. Find mF.

10. In ΔQED, e=60, d=25, mE = 560. Find mD.

11. In ΔRAT, mA = 1480, t=14, a=60. Find mT.

Answers in jumbled order

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-8 1. Consider ΔABC.

a. What is the length of AC?

b. Based on the diagram and your answer to (a), explain where you think the term “tangent” come from.

2. Carefully draw a graph of y = tan(x) on the "restricted domain".

3. Carefully draw a graph of y = tan-1(x).

4. What is the domain and range of the inverse Tangent Function?

5. Find exact value, giving your answer in radians.

a. 1tan 1 b. 1tan 0 c. 1tan 1

6. Find exact value, giving your answer in degrees.

a. 1tan 1 b. 1tan 0 c. 1tan 1

θ

B

C 1

A

Written by Nils Ahbel, Edited by Gloria Barrett © 2005, All rights reserved.

Problem Set 5-9 In 1-4 below, solve the equation in three ways:

a. Find all solutions from 0 to .

b. Solve the equation when 0 2. c. Give the general solution.

1. cos = 0.24

2. cos = -0.24

3. sin = 0.24

4. sin = -0.24

In 5 – 8 below, give the general solution:

5. 3 cos = 2

6. 6 sin - 1 = 0

7. 2 cos = -5

8. sin = -1