Right Triangle Trigonometry

1. Right Triangle Trigonometry By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: December 9, 2010

2. Review of Previous Applications

3. A surveyor is standing 45 feet from the base of a large tree. The surveyor measures the angle of elevation from the ground to the top of the tree to be 67.5o. Find the height of the tree. h 67.5o 45 feet

4. An airplane flying at 4500 feet is on a flight path directly toward an observer. If 30o is the angle of elevation from the observer to the plane, find the distance from the observer to the plane. d 4500 feet 30o

5. In traveling across flat land a driver noticed a mountain directly in front of the car. The angle of elevation to the peek is 4o. After the driver traveled 10 miles, the angle of elevation was 11o. Approximate the height of the mountain. h 11o 4o x 10 mi 10 + x

6. A flagpole at the top of a tall building (and at the edge of the building) is know to be 45 feet tall. If a man standing down the street from the building calculates the angle of elevation to the top of the building to be 55o and the angle of elevation to the top of the flagpole to be 57o. Find the height of the building. 45 h 57o 55o d

7. An observer standing on the cliff adjacent to the ocean looks out and sees an airplane flying directly over a ship. The observer calculates the angle of elevation to the plane to be 14o and the angle of depression to the ship to be 27o. How high above the ship is the airplane if we know that the ship is 1.5 miles from shore? p 14o 1.5 mi 27o b Distance of plane above ship =

8. In Washington, D.C., the Washington Monument is situated between the Capitol and the Lincoln Memorial. A tourist standing at the Lincoln Memorial tilts her head at an angle of 7.491° in order to look up to the top of the Washington Monument. At the same time, another tourist standing at the Capitol steps tilts his head at a 5.463° to also look at the top of the Washington Monument. Find the distance from the Lincoln Memorial to the Washington Monument.

9. Additional Applications

10. A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. d a 54o 20sin(36o) 20 nm 78.181o 20 nmph for 1 hr 36o θ b 20cos(36o) 40 nm 20 nmph for 2 hrs Bearing: N 78.181o W

11. A ship leaves port at noon and heads due west at 20 knots, or 20 nautical miles (nm) per hour. At 2 P.M. the ship changes course to N 54o W. Find the ship’s bearing and distance from the port of departure at 3 P.M. d a 54o 20sin(36o) 20 nm 78.181o 20 nmph for 1 hr 36o θ b 20cos(36o) 40 nm 20 nmph for 2 hrs Bearing: N 78.181o W

12. Two lookout towers are 50 kilometers apart. Tower A is due west of tower B. A roadway connects the two towers. A dinosaur is spotted from each of the towers. The bearing of the dinosaur from A is E 47o N. The bearing of the dinosaur from tower B is W 32o N. Find the distance of the dinosaur to the roadway that connects the two towers. h 47o 32o A B x 50– x

13. h 47o 32o A B x 50– x 19.741 km

14. Two lookout towers spot a fire at the same time. Tower B is Northwest of Tower A. The bearing of the fire from tower A is N 33o W and is calculated to be 45 km from the tower. The bearing of the fire from tower B is N 63o E and is calculated to be 72 km from the tower. Find the distance between the two towers and the bearing from tower A to tower B. a c 63o 72 d B b 45 s 33o 45cos(330) – 72sin(630) b – d A a + c 45sin(330) + 72sin(630)

15. a c 63o 72 d B b 45 s 88.805 km 33o b – d 45cos(330) – 72sin(630) A a + c 45sin(330) + 72sin(630)

16. a c 63o 72 d B b 45 s 88.805 km 33o b – d 45cos(330) – 72sin(630) θ A a + c 45sin(330) + 72sin(630)