Superlesson
Project 10-1

Superlesson
Project 10-2

Superlesson
Project 10-3

Advanced Algebra

Chapter 10, 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.

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. .

d. Find the Radius of the earth (use the mean radius to the nearest kilometer).

e. Use trigonometric ratios to find OM and LM .

f. How long will the tunnel be?

g. How far below the earth's surface will the tunnel run at its deepest point?

h. How much shorter is the tunnel than the distance from Los Angeles to New York along the earth's surface? Explain your method.

 

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?

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.

b. How do you know that the cities form an approximate spherical right triangle? Which city is at the right angle?

c. Use the distances between cities to show that the Pythagorean Theorem is not valid for spherical right triangles.

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.)

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?