Richard P.

asked • 11/01/23

Law of Cosines (Three islands)

The diagram shows three islands in a bay. You rent a boat and plan to visit each of these remote islands. If you are on island​ A, on what bearing should you navigate to go to island​ C? (The shape is a triangle.)


a = 8 miles

b = 5 miles

c = 7 miles


You should navigate on a bearing of _______. (Round to the nearest degree as​ needed.)

Mark M.

There is no diagram!
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11/01/23

Martin C.

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For Mark: It is hard to put a diagram into Ask an Expert. For Richard: What is the bearing of the line from A to B, and what order (clockwise or counterclockwise) are the islands A, B, and C encountered in going around the circumscribed circle through all of them? Even once we find all the angles of triangle ABC, we still won't know what direction to sail in unless we know which way is north. The angle at C can be calculated as the angle whose cosine is (64 + 25 - 49) / 80 = 1 / 2, so angle C = 60 degrees. Similarly, the angle at B = inverse cosine((64 + 49 - 25) / 112) = 38.2 degrees, and the angle at A = inverse cosine((49 + 25 - 64) / 70) = 81.8 degrees. Sure enough, the angles add up to 180 degrees. If we take the line from A to B as being due north, and A, B, and C are in counterclockwise order as you visit them, then we should sail on a heading 81.2 degrees west of north to reach island C.
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11/05/23

1 Expert Answer

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Raymond B. answered • 11/05/23

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8^2=5^2+7^2-2(5)(7)CosA


(64-74)/-70=CosA


angle A=Cos^-1(1/7)= about 81.787 degrees


use law of sines or cosines to find angles B and C


7^2=5^2+8^2-2(40)cosC

cosC= (49-89)/-80=1/2

C= cos^-1(.5)= 60 degrees

B=180-60-81.787= about 38.213 degrees

b^2=a^2+c^2-2acCosB

25=64+49-2(56)CosB

(25-113)/-112=CosB

CosB=88/112=22/28=11/14

B=cos^-1(11/14)= about 38.2132




then put that into your diagram to find the bearing needed to go from point A to C


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