# Thread: Frame of reference question. Polar co-ordinates

1. ## Frame of reference question. Polar co-ordinates

A lighthouse keeper spots a ship sailing away from the shore. She marks the bearing from North to the ship as $\theta$=24 degrees and estimates the distance to the ship as 2.4km. The captain of the ship notes the ship is heading $\phi$ = 30 degrees from north.

If the captain similarly estimates the distance to the lighthouse is 2.4km what are the polar coordinates of the lighthouse in the ships frame of reference?

ok so obviously r= 2.4 but for the angle i keep getting 186 degrees but that's wrong. Can someone explain to me how to get the angle?

2. ## Re: Frame of reference question. Polar co-ordinates

Hello, linalg123!

You need a better sketch.

A lighthouse keeper spots a ship sailing away from the shore.
She marks the bearing from North to the ship as $\theta = 24^o$ and estimates the distance to the ship as 2.4km.
The captain of the ship notes the ship is heading $\phi =30^o$ from north.

If the captain similarly estimates the distance to the lighthouse is 2.4km,
what are the polar coordinates of the lighthouse in the ship's frame of reference?

Code:
              B        C
:       *
:     *
A       :30°*
:       : * 60°
:     S o - - - - - D
:      * 114°
:     *
:    *
:24°* 2.4
:  *
: *
:* 66°
L o - - - - - - - - E
The lighthouse is at $L.$
The ship is at $S:\;\angle ALS = 24^o,\;LS = 24\text{ km}$

The ship is heading to $C.$
$\angle BSC = 30^o,\;\angle CSD = 60^o.$
But none of this is relevant.

We have: $\angle SLE = 66^o \quad\Rightarrow\quad \angle DSL = 114^o$

Using $S$ as the origin, the polar coordinates of $L$ are:
. . . $(2.4,\:\text{-}114^o)\,\text{ or }\,(2.4,\:246^o)$

Recall that in polar coordinates,
. . $\theta$ is measured from the positive x-axis,
. . in this instance, $SD.$