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E4004 Surveying Computations A Two Missing Distances

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Derivation of Formula - Sine Rule Consider a triangle in which the length of one side is known and the 2 angles at each end of this side are known; i.e. A, C and b are known. A C b

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Derivation of Formula - Sine Rule The remaining parts may be calculated by the use of the sine rule A B C b a c

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Derivation of Formula - Sine Rule A B C b a c

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Derivation of Formula - Traverse Consider Traverse ABCD A B C D The line AD can be calculated by a close

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Derivation of Formula - Traverse Suppose the bearings of two lines DE and EA are known but their distances are unknown A B C D E The triangle ADE has two known angles and one known side and the missing parts can be calculated

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Derivation of Formula - Angles All angles can be calculated by subtracting the known bearings A B C D E

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Derivation of Formula - Angles Earlier discussion suggested that the missing distances could be calculated through an application of the sine rule A B C D E Consider finding the sine of angle (EAD)

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Derivation of Formula - Sine of Angles There is a trig identity So but also So From (i) So ……(i)

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Derivation of Formula - Angles A B C D E

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Derivation of Formula Consider triangle ADE A B C D E

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Derivation of Formula Again consider triangle ADE A B C D E

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Summary of Two Missing Distance Formula A B C D E

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The two missing distances need not occur consequtively in the traverse so long as all of the known lines are used to calculate c and c the formula will hold A-B B-C C-D D-E E-F

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9.4 The Law of Cosines Objective To use the law of cosines to find unknown parts of a triangle.

9.4 The Law of Cosines Objective To use the law of cosines to find unknown parts of a triangle.

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