Bearings | Racquel Sanderson

Category Archives: Bearings

June 26, 2025 · 7:22 am

Trigonometry Question

I don’t know where I found this question, but it does require algebra and problem solving (as well as right trig and Pythagoras)

From a point $A$ , a lighthouse is on a bearing of $026^\circ$ T and the top of the light house is at an angle of elevation of $20.25^\circ$ . From a point $B$ , the lighthouse is on a bearing of $296^\cric$ T and the top of the lighthouse is at angle of elevation of $10.2^\circ$ . If $A$ and $B$ are 500 metres apart, find the height of the lighthouse.

Let’s draw a diagram.

Let the height of the lighthouse be $h$

We can find the angle between $A$ , the lighthouse, and $B$ by using the base triangle

The red line from $L$ is parallel to the two north lines. Hence $\theta=26^\circ+64^\circ=90^\circ$ (Alternate angles in parallel lines are congruent)

It’s a right triangle so we know

(1) $\begin{equation*}500^2=(AL)^2+(BL)^2\end{equation*}$

We are going to use the other two triangles to find $AL$ and $BL$

Substitute $AL$ and $BL$ into equation $1$

$\begin{equation*}500^2=(\frac{h}{tan(20.25)})^2+(\frac{h}{tan(10.2)})^2\end{equation}$

Solve for $h$ .

$\begin{equation*}500^2=7.35h^2+30.89h^2=38.24h^2\end{equation}$

$\begin{equation*}h^2=6538.3\end{equation}$

$\begin{equation*}h=80.9m\end{equation}$

Non-Right Trigonometry Problem

I worked on this question with one of my students (I don’t know where it is from).

Mike leaves the rose bush he was examining and walks 35m in the direction
S $20^{\circ}$ W towards a pond.
From there he walks 70m towards a rotunda. Mike is now 100m from the rose bush.
Find the bearing of the rotunda from the pond.

Let’s try to draw a diagram

Because we don’t the direction Mike walked from the pond, I have drawn a circle with radius 70m centred at the pond.

We know Mike is now 100m from the rose bush. As we don’t know the direction, I have drawn another circle with radius 100m centred at the rose bush. Where the two circles intersect are the possible locations of the rotunda.

First Position

Use the cosine rule to find the angle
$cos\theta=\frac{b^2+c^2-a^2}{2bc}$
$cos\theta=\frac{70^2+35^2-100^2}{2(70)(35)}$
$\theta=cos^{-1}(\frac{-3875}{5250})$
$\theta=137.6^{\circ}$

Using the fact that alternate angles in parallel lines are congruent, we can see
that the bearing from the pond to the rotunda is
$360-(137.6-20)=242.2^{\circ} T$