k.a.n asked in Science & MathematicsMathematics · 8 years ago

# How do you solve this trigonometry word problem with bearing?

A ship is sailing due south. At point R, the watchman observes an iceberg S75 degreesE. After the ship has sailed 7 nautical miles to point O, the bearing to the berg is N45 degreesW. IF the iceberg is point D(and does not move), find the distance RD and OD.

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• 8 years ago

You clearly have an error in your question. With the ship traveling due south, the direction east or west will not change. The iceberg will still be on the same side at point O as it was at point R. So, let's assume that your second bearing has a typo and it is supposed to say "N45 degrees E" putting it on the eastward side like at point R, S75 degrees E. If we make this assumption then the question makes sense. The answer is relatively straightforward.

A ship is sailing due south. At point R, the watchman observes an iceberg S75 degreesE.

After the ship has sailed 7 nautical miles to point O, the bearing to the berg is N45 degreesE.

IF the iceberg is point D(and does not move), find the distance RD and OD.

You end up with a triangle roughly like this (but obviously with different angles).

R-------D

|

|

|

O

So, what do we know?

With D relative to R at S75E we know that the interior angle of R is 75 degrees.

With D relative to O at N45E we know that the interior angle of O is 45 degrees.

We also know that the length of RO is 7 nautical miles.

The interior angles of any triangle add up to 180 degrees. So 180 - (75 + 45) = 60. The interior angle of D must be 60 degrees.

This would be easiest to solve with the Law of Sines. http://en.wikipedia.org/wiki/Law_of_sines

a / sin A = b / sin B = c / sin C

The one side we know is RO so the angle we need is D which is 60 degrees.

So, RO / sin D

7 / sin 60 degrees

7 / 0.866 = 8.083

So, 8.083 must equal b / sin B and c / sin C

or, 8.083 = RD / sin O = OD / sin R

8.083 = OD / sin R

8.083 = OD / sin 75 degrees

8.083 = OD / 0.966

0.966 * 8.083 = OD

8.083 = RD / sin O

8.083 = RD / sin 45 degrees

• Anonymous
8 years ago

First draw a picture and label points. The important thing to remember is that what S75 degrees east means first look straight due south and then go 75 degrees east. If that's confusing, watch this short video for visualization:

Using that information, when I drew my picture, I got angle DRO to be 15 degrees, angle DOR to be 45 and angle RDO to be 120 degrees. From here use the law of sines:

angle RDO will correspond to the 7 miles traveled because the boat is moving past the iceberg. First, we'll solve RD. According to the law of sines:

(RD)/(sin(45))=(7/(sin(120)) --> multiply both sides by sin(45) and get:

(sin(45))/sin(120)*7=RD

5.72=RD if you want an exact answer, the most simplified it gets is (7/3)sqrt(6) after simplifying from sin(120)=sqrt(3)/2 and sin(45)=sqrt(2)/2

Use law of sines again to find OD.

OD/sin15=7/sin120 --> multiply by sin15

OD=7sin15/sin120=2.09 -->if you want exact answer sin15=(sqrt(6)-sqrt(2))/4 by using subtractive angle theorem because sin(15)=sin(45-30), so the exact answer comes out to be (7)(sqrt(6)-sqrt(2)) all divided by 2sqrt(3)