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Sine Rule (Bearings) Trigonometry

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**1. In the examples which follow use the sine rule**

to find the required side or angle. 1. Consider two radar stations Alpha and Beta. Alpha is 140 miles west of Beta. The bearing of an aero plane from Alpha is 032° and from Beta it is 316°. How far is the aero plane from each of the radar stations? Two ports Dundee and Stonehaven are 54 miles apart with Dundee approximately due south of Stonehaven. The bearing of a ship at sea from Stonehaven is 098° and from Dundee it is 048°. How far is the ship from Dundee?

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A ship is sailing north. At noon its bearing from a lighthouse is 240°. Five hours later the ship is 84km further north and its new bearing from the lighthouse is 290°. How far is the ship from the lighthouse now? Two ports P and Q are 35 miles apart with P due east of Q. The bearings of a ship from P and Q are 190° and 126° respectively. a) What is the bearing of port Q from the ship? b) How far is the ship from port P The bearing of an aero plane from Aberdeen is 068° and from Dundee it is 030°. Dundee is 69 miles south of Aberdeen. How far is the aero plane from Aberdeen?

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**Two oil rigs are 80 miles apart with rig**

B being 80 miles east of rig A. The bearing of a ship from rig B is 194° and from A the bearing is 140°. Find the distance of the ship from rig A. At noon a ship lies on a bearing 070° from Dundee. The ship sails a distance of 45 miles south and at 3pm its new bearing from Dundee is 130°. How far is the ship from Dundee now? If the ship maintains the same speed, how long will it take to return to Dundee? An aero plane leaves Leuchers airport and flies 80 miles north. It then changes direction and flies 120 miles east. The plane now turns onto a bearing of 126° and flies a further 164 miles. Calculate the bearing and distance of the plane from Leuchers.

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**Question Solution 1 122.4 miles ; 103.8 miles 2 69.8 miles 3**

95.0 kilometers 4 Bearing 306° ; 22.9 miles 5 56.0 miles 6 114.9 miles 7 48.8 miles ; 3 hours 15 minutes 8 236.2 miles ; bearing 269° Note: the last question requires the cosine rule as well as the sine rule.

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