Radio Coverage

Part of the Triangles Without Right Angles section of Coddy's Trigonometry journey. Lesson 14 of 44.

A straight road leaves a town A. A radio mast M stands off the road, and its signal reaches a fixed distance. A car on the road enters the signal at one point and leaves it at another: two points P at the same distance from M. That is the ambiguous case on a map.

$A road from A with two points P₁ and P₂ at the same distance 10 from the mast M$

Say the mast is 16 km from A, the road makes 25° with the line AM and the range is 10 km. At a point P where the signal starts or stops, sin(P)/16 = sin(25°)/10, so sin P is about 0.6762 and the angle at P is about 42.55° or 137.45°. Both leave room for 25°, so the road crosses the edge of the signal twice.

Challenge

A road leaves town A at 32° to the line from A to a radio mast M, which is 18 km from A. The signal reaches 12 km. At each point P where a car on the road enters or leaves the signal, find the angle θ at P, to two decimal places, and press Check.

Try it yourself

$Use the figure to find theta$