In a previous post, I introduced bearing and distances. Now, let's solve some bearing and distances questions. The following are the steps we will take in solving the problems:

* Sketch the bearing
* Sketch out a triangle
* Apply the sine or cosine rule (or both).


Example 3: Three towns, A, B and C are situated so that |AB| = 60km and |AC| = 100km. The bearing of B from A is 060degree and the bearing of C from A is 290degree. Calculate:

a) the distance |BC|,

b) the bearing of B from C

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Solution
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a) The distance |BC|

From the question, we notice that the two bearings provided are from A, so it will be wise to start the sketch from point A, below is a sketch of B from A:




Then of C from A,



[ Notice that the lenght of the lines in the sketch reflects thats 100km is 'longer' that 60km ]

At point A, the remaining unknown angle is given by 360degree - 290degree = 70degree.

So the sketch becomes



The triangle ABC is then given by



Using the cosine rule to find the length a;

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

a^2 = 100^2 + 60^2 - 2(60 X 100)Cos 130

a^2 = 10000 + 3600 - 12000Cos 130

From a calculator or tables of values, cos 130 = - 0.6428,then:

a^2 = 13600 - 12000 X -0.6428

a^2 = 13600 + 7713.6

a^2 = 21313.6

a = √21313.6

a = 145.99

a = 146 km ( coverted to 1 decimal places )

Therefore, the distance |BC| = 146 km


b) The bearing of B from C



As shown in the diagram above, the bearing of B from C is given by 0xdegree in three figure bearings. To get the value of x, we will first find the value of the internal angle C in the triangle by using the sine rule.

c/sin C= a/sin A

Making sinC the subject of the formular gives;

sin C = (c X sin A)/a

sin C = (60 X Sin 130)/146


sin C = (60 X 0.7660)/146


sin C = 45.96/146

sin C = 0.3148

We will find the 'arc sin' of 0.3148

C = sin^{-1}0.3148

C = 18.35degree

Remember the angle C we just found the value, is not the bearing of B from C, it is the internal angle of the triangle at C, see the sketch below:



From the diagram above, we see that;

x = 180 - (70 + 18.25) = 91.65degree

x = 91.7degree.

Whablamo! The bearing of B from C is 091.7 degree.

Happy math.

See full post here http://samimath.com/bearing-distance-explained-examples/


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