The question is an illustration of bearing (i.e. angles) and distance (i.e. lengths)
The distance between both lighthouses is 5783.96 m
I've added an attachment that represents the scenario.
From the attachment, we have:
[tex]\mathbf{\angle A = 180^o - 120^o\ 43'}[/tex]
Convert to degrees
[tex]\mathbf{\angle A = 180^o - (120^o +\frac{43}{60}^o)}[/tex]
[tex]\mathbf{\angle A = 180^o - (120^o +0.717^o)}[/tex]
[tex]\mathbf{\angle A = 180^o - (120.717^o)}[/tex]
[tex]\mathbf{\angle A = 59.283^o}[/tex]
[tex]\mathbf{\angle B = 39^o43'}[/tex]
Convert to degrees
[tex]\mathbf{\angle B = 39^o + \frac{43}{60}^o}[/tex]
[tex]\mathbf{\angle B = 39^o + 0.717^o}[/tex]
[tex]\mathbf{\angle B = 39.717^o}[/tex]
So, the measure of angle S is:
[tex]\mathbf{\angle S = 180 - \angle A - \angle B}[/tex] ---- Sum of angles in a triangle
[tex]\mathbf{\angle S = 180 - 59.283 - 39.717}[/tex]
[tex]\mathbf{\angle S = 81}[/tex]
The required distance is distance AB
This is calculated using the following sine formula:
[tex]\mathbf{\frac{AB}{\sin(S)} = \frac{AS}{\sin(B)} }[/tex]
Where:
[tex]\mathbf{AS = 3742}[/tex]
So, we have:
[tex]\mathbf{\frac{AB}{\sin(81)} = \frac{3742}{\sin(39.717)}}[/tex]
Make AB the subject
[tex]\mathbf{AB= \frac{3742}{\sin(39.717)} \times \sin(81)}[/tex]
[tex]\mathbf{AB= 5783.96}[/tex]
Hence, the distance between both lighthouses is 5783.96 m
Read more about bearing and distance at:
https://brainly.com/question/19017345
