The Law of Sines and Cosines and Its Applications

An architect’s client wants to build a home based on the architect Jon Lautner’s Sheats-Goldstein House. The length of the patio will be 60 feet. The left side of the roof will be at a 49º angle of elevation, and the right side will be at an 18º angle of elevation. Determine the lengths of the left and right sides of the roof and the angle at which they will meet.

(a)  

For the initial 90 miles of a flight, the pilot heads 8º off course in order to avoid a storm. The pilot then changes direction to head toward the destination for the remainder of the flight, making a 157 angle to the first flight course. Determine the distance of a direct flight to the destination

• 114.9 miles.

• 125.7 miles.

• 135.9 miles.

• 136.1 miles.

A ski lift rises at a 28 angle during the first 20 feet up a mountain to achieve a height of 25 feet, which is the height maintained during the remainder of the ride up the mountain. Determine the length of cable needed for this initial rise.

(a)  

During her shift, a pilot flies from the Columbus to Atlanta, a distance of 448 miles, and then on to the Phoenix, a distance of 1583 miles. From Phoenix, she returns home to Columbus, a distance of 1667 miles. Determine the angles of the triangle created by her flight path.

• 15.6 , 71.5 , and 92.9

• 15.6 , 71.6 , and 92.8

• 14.8 , 72.5 , and 92.7

• 15.7 , 69.4 , and 94.9

A pilot flies her plane on a heading of 35° from point 𝑋𝑋 to point 𝑌𝑌, which is 400 mi from 𝑋𝑋. Then she turns and flies on a heading of 145° to point 𝑍𝑍, which is 400 mi from her starting point 𝑋𝑋. What is the heading of 𝑍𝑍 from 𝑋𝑋, and what is the distance 𝑌𝑌?

(a)  

A ship is sailing due north. At a certain point, the captain of the ship notices a lighthouse 12.5 km away from the ship, at the bearing of 𝑵38.8°𝑬. Later on, the bearing of the lighthouse becomes S44.2°𝑬. In meters, how far did the ship travel between the two observations of the lighthouse?

• 1.181km

• 1.175km

• 1.178km

• 1.168km