Advanced Algebra
Chapter 10 Answers
Trigonometry
Trigonometric ratios have wide applications in spherical geometry. In the following problems, you'll use trigonometric ratios to find distances and angles on the earth's surface, beneath the surface, and thousands of miles above the surface.
Part A, Triangle Trigonometry
1. An engineer has proposed building a tunnel directly from Los Angeles
to New York City 
to house a high-speed train between the cities.
Officials want to know how long the tunnel will be and how far below the
earth's surface the tunnel will run at its deepest point.
a. Find the latitude and longitude of Los Angeles, L( a, b), and New York, N(c, d), at the U.S. Gazetteer Web site.
[LA: 34.09095° N, 118.40844° W; NY: 40.77436° N, 73.97213° W]b. If two points on the earth's surface have latitudes and longitudes (a, b) and (c, d), the cosine of the central angle determined by the points is given by sine a sine c + cos a cos c cos [d b]). Find the cosine of central angle
. Use it to find m
. (Hint: Use the "inverse cosine" keys
on your calculator.)
[]
c..
[17.75°]d. Find the Radius of the earth (use the mean radius to the nearest kilometer).
[6371 km]e. Use trigonometric ratios to find OM and LM .
[OM = 6068 km; LM = 1942 km]f. How long will the tunnel be?
[2 x LM = 3884 km]g. How far below the earth's surface will the tunnel run at its deepest point?
[OA OM = 6371 6068 = 303 km]h. How much shorter is the tunnel than the distance from Los Angeles to New York along the earth's surface? Explain your method.
[
arc length LAN = 2(3.14)(6371)(35.5)/360) ≈ 3945 km
arc length LAN tunnel length ≈ 3945 3884 = 61 km]
Part C, Arc Length and Sector Area
2.Read about Geosynchronous satellites.
a. At what height (km) above the earth's surface do these satellites orbit?
[35,786 km]b. Find the angular speed (radians/day), radius of orbit (km), and linear speed (km/day) for a geosynchronous satellite.
Part E, Making Connections
3. 
is a spherical right triangle with central
angles measuring a, b, and c, respectively, and right
angle C.
a. Find the latitudes and longitudes of Panama City, Panama, Clarion, PA, and Ogden, UT, and the distances (km) between each pair of cities at the Distances Web site.
[Panama City, 9° 04'01" N, 79° 22' 59" W
Clarion, 41° 12' 37" N,. 79° 22' 50" W
Ogden, 41° 13' 40" N, 111° 57' 57" W
PC = 3561 km, CO = 2716 km, OP = 4791 km]b. How do you know that the cities form an approximate spherical right triangle? Which city is at the right angle?
[Panama City and Clarion are on the same line of longitude. Clarion and Ogden are on the same line of latitude. Lines of longitude and latitude are perpendicular. Clarion is at the right angle.]c. Use the distances between cities to show that the Pythagorean Theorem is not valid for spherical right triangles.
[35612. + 27162 ≠ 47912]d. Assume that the earth is a sphere with great circles of circumference of 40,000 km. Find a, b, and c, the measures of the central angles, to the nearest tenth of a degree. (See figure of the earth above.)
[32.0°, 24.4°, 43.1°]e. The formula cos a cos b = cos c is called the Pythagorean Theorem for spherical right triangles. Is the formula valid for the Panama City-Clarion-Ogden triangle? What assumption did you make that might explain your result?
[cos 32.0° cos 24.4° cos 43.1°
0.8480 (0.9107) 0.7302
0.7723 ≈ 0.7302
The right and left sides of the formula generate values that are close but not equal. The assumption that best explains the discrepancy is that the earth is not a sphere. Rather, it is flattened at the poles due to its rotation. The flattening produces great circles of differing lengths depending on where they are drawn. In particular, the great circle containing Panama City and Clarion, a line of longitude passing through both the North and South (flattened) poles, is shorter than the great circles containing the other pairs of cities.]
For surveyors, the ability to measure distances accurately is a practical matter. For vulcanologists--scientists who study volcanoes--it can be a matter of life and death.
Part A, Law of Cosines
1. Use the Volcanic Deformation Project Web site to answer the following questions.
a. What use do vulcanologists in the Volcano Deformation Project make of EDM (electronic distance measurement)?
[They use EDM to measure distance changes to points on a volcano caused by "deformation"--bulges in the volcano created by the build-up of pressure from within. Deformation is a precursor to eruption.]b. The volcano Mount St. Helens in the state of Washington erupted in 1980. View the slide show of the eruption at this Web site. Suppose that on May 17, 1980, a vulcanologist used EDM to measure the distance to the nearest point on the bulge described in Slide #3. How much more quickly would a laser pulse make the round trip from the EDM instrument to the point on the bulge and back again than it would have before the bulge began to appear? (A laser beam travels at the speed of light, 186,282.3976 mi/sec.) Express your answer in scientific notation.
[9 x107 sec]c. View the lava dome in Slide #23. Find lengths AC and BC (ft). (Assume that the width given as "nearly" is exact.)
[AC = 3300 ft; BC = 535 ft]d. Find
.
[9.2°]e. A vulcanologist at A made EDM measurements of a lava dome as shown. Use the Law of Cosines to find BD. Then find the height of the dome.
[BD = 825 ft; BC = 401 ft]
Part C, Making Connections
2.View the before and after photos of Mount St. Helens taken from nearby Spirit Lake at this Web site.
a. What are the elevations of the volcano before and after the eruption?
[before: 9677 ft; after: 8364 ft]b. From the top of the "old" Mount St. Helens, the angle of depression of the spot at Spirit Lake where the photos were taken was 16.41°. From the top of the "new" mountain, the angle is 13.95°. Use the Law of Sines and the difference in elevations between the old and new mountains to find length SB (ft).
[29,344 ft]c. Find BC, the height of Mount St. Helens above Spirit Lake today.
[7074 ft]d. Find the elevation of Spirit Lake above sea level.
[1290 ft]
Functions associated with the geometry of the earth often are periodic, and often can be expressed using trigonometry.
Part A, Period and Amplitude
1. As the Mayan Indians discovered, sunrise is a periodic function that depends on the time of year and one's location on the globe. You can find sunrise times for any year and for most locations in the United States by logging onto the U.S. Naval Observatory's Web site.
a. Enter a year, "sunrise/sunset" table, your state, and your city or a city near where you live. Then click on "Compute Table."
[Tables will vary depending on year and location.]b. Record sunrise time for January 1, January 15, January 29, February 12, and every two weeks thereafter for the entire year.
[Times will vary.]c. Calculate the time that is midway between the earliest and latest times in your data.
[Times will vary.]d. Graph your data. Record dates on the horizontal axis. Record times on the vertical axis. Use the midway time that you calculated in c as the zero point of your vertical axis.
[Graphs will vary but should look like sine or cosine curves. The maximum value should occur in late June, the minimum value in late December, and zero values in late March and late September.]e. What is the amplitude of the function you graphed? What is the period?
[The amplitude is the difference between the maximum or minimum value and the zero value. The period is one year.]
Part C, Making Connections
2. The length of a nautical mile is based on the circumference of a great circle on the earth's surface. Because the earth is slightly egg-shaped rather than spherical, this length varies.
a. Find the length of a nautical mile at the equator and on the "official nautical mile" Great circle Web site.
[6087.15 ft; 6076.1 ft]b. Find the length (ft) of the shortest nautical mile, the one based on the circumference of the great circle through the poles (24,817 mi).
[6066.4 ft]Use for c-g
From 0° latitude (the equator) to 90° latitude (the North or South pole), the length of a nautical mile can be modeled by the cosine function. Find each:
c. maximum value
[6087.15]d. minimum value
[6066.4]e. average y-value
[6087.15 + 6066.4/2) ≈ 6076.775]f. amplitude
[6087.15 6076.775 = 10.375]g. period
g. phase shift
[0]h. What is the equation of the function?
[y = 6076.775 + 10.375 cosine 2x]i. Find the latitude of Your city at this Web site. Then calculate the length of a nautical mile at your latitude.
[Lengths will vary. Students should substitute their latitudes for x in the equation in h.]
