Radio direction finders are placed at points A and B, which are 4.44 mi apart on an east-west line, with A west of B. The transmitter has bearings 39.3 degrees from A and 313.9 degrees from B. Find the distance from A.

Given one distance and two angles, we will need to use the Law of Sines. For this, we need to know the internal angles of the triangle formed by the various bearing lines.

The angle between the bearings of A and B from the transmitter will be the difference of the reverse of the given bearings.

A from T = 39.3° +180° = 219.3°

B from T = 313.9° -180° = 133.9°

Then the angle at T between receivers is ...

219.3° -133.9° = 85.4°

The angle between A and T as measured at B will be ...

313.9° -270° = 43.9°

These angles and length AB can be used with the Law of Sines to find AT:

AT/sin(B) = AB/sin(T)

AT = AB(sin(B)/sin(T)) = (4.44 mi)·sin(43.9°)/sin(85.4°)

AT ≈ 3.09 mi

The distance of the transmitter from A is about 3.09 miles.

Radio direction finders are placed at points A and​ B, which are 4.44 mi apart on an east-west ​line, with A west of B. The transmitter has bearings 39.3 degrees from A and 313.9 degrees from B. Find the distance from A.

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Radio direction finders are placed at points A and B, which are 4.44 mi apart on an east-west line, with A west of B. The transmitter has bearings 39.3 degrees from A and 313.9 degrees from B. Find the distance from A.