Trigonometry : Bearings and Directions

1.        Two ships leave a port at the same time. The first ship sails on a bearing of $ \displaystyle 032{}^\circ $ at $ \displaystyle 16$ km/h and the second on a bearing of $ \displaystyle 122{}^\circ $ at $ \displaystyle 24$ km/h. How far apart are they after $ \displaystyle 2.5$ hours?

2.       The bearing from point $ \displaystyle A$ to point $ \displaystyle B$ is $ \displaystyle S\ 55{}^\circ E$ and from a point $ \displaystyle B$ to point $ \displaystyle C$ is $ \displaystyle N\ 35{}^\circ E$. If a ship sails from $ \displaystyle A$ to $ \displaystyle B$, a distance of $ \displaystyle 81$ km, and from $ \displaystyle B$ to $ \displaystyle C$, a distance of $ \displaystyle 74$ km, how far is it from $ \displaystyle A$ to $ \displaystyle C$?

3.        A pilot flew $ \displaystyle 320$ miles at $ \displaystyle 068{}^\circ 1{0}'$, then $ \displaystyle 540$ miles at $ \displaystyle 158{}^\circ 1{0}'$. The pilot then flew back to his starting point. Find the length of the last flight.

4.        A ship leaves a harbour on a bearing of $ \displaystyle S\ 55{}^\circ E$ and travels for $ \displaystyle 120$ km. It then turns and continues on a bearing of $ \displaystyle N\ 35{}^\circ E$ for $ \displaystyle 62$ km. How far is the ship from the harbour.

5.        A boy could have walked along a straight road that heads $ \displaystyle N\ 30{}^\circ 3{7}' E$, but, instead, he walked due east $ \displaystyle 153.9$ yards and then due north to the end of the road. How far out of his way did the boy walk?

How far out of his way = (လမ်းအတိုင်းသွားရင် လျှောက်ရမယ့် အကွာအဝေးထက်) ဘယ်လောက် ပိုလျှောက်ခဲ့ရလဲ

6.        Two fishing boats start from a port. One travels $ \displaystyle 𝟏𝟓$ nautical miles per hour on a bearing of $ \displaystyle 025{}^\circ$ and the other travels $ \displaystyle 𝟏𝟖$ nautical miles per hour on a bearing of $ \displaystyle 𝟏𝟎𝟎°$. Assuming each maintains its course and speed, how far apart will the fishing boats be after two hours?

7.        A plane leaves from Yangon International Airport and travels due west at $ \displaystyle 570$ mi/hr. Another plane leaves $ \displaystyle 20$ minutes later and travels $ \displaystyle 22{}^\circ$ west of north at the rate of 585 mi/h. How far apart are they 40 minutes after the second plane leaves.

8.        An airplane travels on a bearing of $ \displaystyle 200{}^\circ$ for $ \displaystyle 𝟏𝟓𝟎𝟎$ miles and then changes to a bearing of $ \displaystyle 250{}^\circ$ and travels an additional $ \displaystyle 𝟓𝟎𝟎$ miles. How far is the airplane from its starting point?

9.        Two runners start from the same point. The first one heading due north at a constant speed of $ \displaystyle 12$ km/h. After $ \displaystyle 2.5$ hours, the second runner is at a bearing of $ \displaystyle 110{}^\circ$ from his starting point and 166.8 km away from the first.

(a) How far is the second runner from the starting point?

(b) How fast is the second runner runs on average?

10.       Yoon Yoon lives $ \displaystyle 𝟐$ miles west of her school and her friend May Kyaw lives $ \displaystyle 𝟑$ miles directly northeast of her.

(a) Draw a diagram representing this situation.

(b) How far does May Kyaw live from school? (c) What is the direction of school from May Kyaw's house?

11.       The tallest trees in the world are taller than a football field is long. Find the height of one of these trees, given the information in the figure.

12.       Two lighthouses are $ \displaystyle 𝟑𝟎$ miles apart on each side of shorelines running north and south, as shown. Each lighthouse keeper spots a boat in the distance. One lighthouse keeper notes the location of the boat as $ \displaystyle 40{}^\circ $ east of south, and the other lighthouse keeper marks the boat as $ \displaystyle 32{}^\circ $ west of south. What is the distance from the boat to each of the lighthouses at the time it was spotted?

13.       Two radar stations located $ \displaystyle 20$ miles apart both detect a UFO between them. The angle of elevation measured by the first station is $ \displaystyle 35$ degrees. The angle of elevation measured by the second station is $ \displaystyle 15$ degrees. What is the altitude of the UFO?

UFO = undefined flying object = ပန်းကန်ပြားပျံ
radar stations = ရေဒါစခန်းများ

14.       To measure the height of a hill, a surveyour measures the angle of elevation to the top of the hill to be $ \displaystyle 24$ degrees. He then moves back $ \displaystyle 200$ feet and measures the angle of elevation to be $ \displaystyle 22$ degrees. Find the height of the hill.

15.       A bridge is built across a small lake from a gazebo to a dock. The bearing from the gazebo to the dock is $ \displaystyle S\ 41{}^\circ W$. From a tree $ \displaystyle 100$ meters from the gazebo, the bearings to the gazebo and the dock are $ \displaystyle S\ 74{}^\circ E$ and $ \displaystyle S\ 28{}^\circ E$, respectively. Find the distance from the gazebo to the dock.

Gazebo = အပန်းဖြေစံအိမ်

16.       A boat is traveling due east parallel to the shoreline at a speed of $ \displaystyle 10$ miles per hour. At a given time, the bearing to a lighthouse is $ \displaystyle S\ 70{}^\circ E$, and $ \displaystyle 15$ minutes later the bearing is $ \displaystyle S\ 63{}^\circ E$. The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline?

shoreline = ကမ်းပါး

17.       On a small lake, a boy swims from point of $ \displaystyle A$ to point of $ \displaystyle B$ at a bearing of $ \displaystyle N\ 28{}^\circ E$, then to point of $ \displaystyle C$ at a bearing of $ \displaystyle N\ 58{}^\circ W$, and finally back to point of $ \displaystyle A$, as shown in the figure below. Point of $ \displaystyle C$ lies of $ \displaystyle 800$ meters directly north of point of $ \displaystyle A$. Find the total distance that a boy swims.