test question (1 Viewer)

[mathsbrain]

Ships A and B leave port P at the same time.
Ship A sails 4km on a bearing of 40 degrees , then a further 6km on a bearing of 155 degrees to reach destination D.
Ship B sails directly from P to the the destination D.
In which direction does Ship B sail?

I solved this in two ways (sine rule and cosine rule) and i got two different answers(but i have a gut feeling that cosine rule has the right answer)

Anyways, applying sine rule(sinx/6=sin65/5.63), gives two answers using ambiguous case, namely 75 degrees and 105 degrees
Cosine rule only gives 75 degrees(cosx=(4^2+5.63^2-6^2)/(2*4*5.63)).
Can someone please explain why the answer is only 75 degrees?

 

[mathsbrain]

can anyone help?

 

[Drongoski]

Ship B was sailing in the direction: 114.93 deg



PD = 5.6315 .... km.

 

[mathsbrain]

Ship B was sailing in the direction: 114.93 deg



PD = 5.6315 .... km.

Hi yes i have gotten the answers, but was just wondering, if you use the sine rule, you get two answers 75 degrees and 105 degrees by ambiguous case, whereas cosine rule gives only 75 degrees. can you explain the reason for rejecting 105 degrees?

 

[Drongoski]

Hi yes i have gotten the answers, but was just wondering, if you use the sine rule, you get two answers 75 degrees and 105 degrees by ambiguous case, whereas cosine rule gives only 75 degrees. can you explain the reason for rejecting 105 degrees?

I used the cosine rule to work out the length PD. With a diagram of the situation, there is no Sine Rule ambiguity at all.

 

[bujolover]

Ships A and B leave port P at the same time.
Ship A sails 4km on a bearing of 40 degrees , then a further 6km on a bearing of 155 degrees to reach destination D.
Ship B sails directly from P to the the destination D.
In which direction does Ship B sail?

I solved this in two ways (sine rule and cosine rule) and i got two different answers(but i have a gut feeling that cosine rule has the right answer)

Anyways, applying sine rule(sinx/6=sin65/5.63), gives two answers using ambiguous case, namely 75 degrees and 105 degrees
Cosine rule only gives 75 degrees(cosx=(4^2+5.63^2-6^2)/(2*4*5.63)).
Can someone please explain why the answer is only 75 degrees?

In such a situation, you would consider 75^o to be correct, since it's the common answer when you use both the sine and cosine rules.

As for why 105^o can't be an answer, well, I can't think of a particularly good reason. The best I can think of is subbing cos105^o into the cosine rule as follows and seeing if it satisfies it:
Let's try and prove that 6² = 4² + 5.631547162² - 2*4*5.631547162cos105^o
Solve both sides: obviously the LHS = 36, but the RHS gets 59.42864687, therefore, LHS ≠ RHS, and the angle hence can't be 105^o, but only 75^o.

Hope this helps.

 

[mathsbrain]

In such a situation, you would consider 75^o to be correct, since it's the common answer when you use both the sine and cosine rules.

As for why 105^o can't be an answer, well, I can't think of a particularly good reason. The best I can think of is subbing cos105^o into the cosine rule as follows and seeing if it satisfies it:
Let's try and prove that 6² = 4² + 5.631547162² - 2*4*5.631547162cos105^o
Solve both sides: obviously the LHS = 36, but the RHS gets 59.42864687, therefore, LHS ≠ RHS, and the angle hence can't be 105^o, but only 75^o.

Hope this helps.

thanks thats very clever. appreciate your help!

 

[bujolover]

thanks thats very clever. appreciate your help!

No worries.