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 $\displaystyle \theta$=24 degrees and estimates the distance to the ship as 2.4km. The captain of the ship notes the ship is heading $\displaystyle \phi$ = 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?

Re: Frame of reference question. Polar co-ordinates

Hello, linalg123!

You need a better sketch.

Quote:

A lighthouse keeper spots a ship sailing away from the shore.

She marks the bearing from North to the ship as $\displaystyle \theta = 24^o$ and estimates the distance to the ship as 2.4km.

The captain of the ship notes the ship is heading $\displaystyle \phi =30^o$ 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 $\displaystyle L.$

The ship is at $\displaystyle S:\;\angle ALS = 24^o,\;LS = 24\text{ km}$

The ship is heading to $\displaystyle C.$

$\displaystyle \angle BSC = 30^o,\;\angle CSD = 60^o.$

But none of this is relevant.

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

Using $\displaystyle S$ as the origin, the polar coordinates of $\displaystyle L$ are:

. . . $\displaystyle (2.4,\:\text{-}114^o)\,\text{ or }\,(2.4,\:246^o)$

Recall that in polar coordinates,

. . $\displaystyle \theta$ is measured from the *positive x-axis,*

. . in this instance, $\displaystyle SD.$