Question Type 2: Exploring complex formulations using bearings and angles Exercises

From a lighthouse at $L$, bearings to two ships $A$ and $B$ are $060^{\circ}$ and $150^{\circ}$ respectively. If the ships are each $20\text{ km}$ from $L$, calculate the distance $AB$.

A patrol boat moves from $P$ to $Q$ on a bearing of $135^\circ$ for $25$ km, then to $R$ on a bearing of $015^\circ$ for $30$ km.

Determine the bearing of $R$ from $P$.

From station $X$, an aircraft is sighted on a bearing of $070^\circ$. Another station $Y$, $30 \text{ km}$ due east of $X$, observes the same aircraft on a bearing of $310^\circ$. Determine the position of the aircraft relative to $X$ by coordinates $(x, y)$.

An airplane departs Q, flies 200 km on a bearing of 350°, then flies 150 km on a bearing of 260°, and finally returns directly to Q. Calculate the bearing of this return leg.

An aircraft flies from D to E on a bearing of 220° for 150 km, then turns onto a bearing of 030° and flies a further 180 km to F. Calculate the straight-line distance DF, giving your answer to the nearest kilometre.

Stations A and B are 40 km apart on a due east-west line (B east of A). A ship S is such that its bearing from A is $060^\circ$ and from B is $330^\circ$. Determine the distance AS.

A navigator at $N$ hears two bearings to a drifting buoy: $020^{\circ}$ and, after moving $15\text{ km}$ due north, $050^{\circ}$. Find the position of the buoy relative to the initial point $N$.

A plane travels from R to S on a bearing of $315^\circ$ for $120$ km, then turns through $150^\circ$ clockwise and flies $80$ km to T. Find the bearing of T from R, giving your answer to the nearest degree.

Three survey points $A$, $B$ and $C$ form a triangle. The bearing of $B$ from $A$ is $045^\circ$, and the bearing of $C$ from $A$ is $120^\circ$. Given $AB=5 \text{ km}$ and $AC=7 \text{ km}$, calculate the angle at $A$ in the triangle $ABC$.

A ship travels from $P$ to $Q$ on a bearing of $145^\circ$ for $35 \text{ km}$. Find the coordinates of $Q$ if $P$ is at $(10, -5)$, using north as the positive $y$-axis and east as the positive $x$-axis. Round your answers to one decimal place.

From point $M$, the bearings of two landmarks $C$ and $D$ are $012^\circ$ and $270^\circ$ respectively. If $MC = 12\text{ km}$ and $MD = 8\text{ km}$, find the distance $CD$ to the nearest kilometre.

Points A and B have coordinates A(2, 1) and B(8, 5). Calculate the bearing of B from A to the nearest degree.