Trig Right triangle Trigonometry

[TheKracken]

Homework Statement

As ship is anchor off a long straight shoreline that runs north and south. From twi observation points. 15 miles apart on shore the bearings of the ship are N 31 ° E and S 53 ° E. What is the shortest distance from the ship to the shore.

Homework Equations

Sin θ Opp/ Hyp
Cos θ Adj/ Hyp
Tan θ Opp/ Adj

The Attempt at a Solution

I have drawn out the photo and it has a base of 15 miles and degree angles of 31, 53 and 96
I assumed the fastest route was a straight line back and that would cut the triangle into 2 different right triangles with their bases added together = 15 miles. I am confused if I should do a system of linear equations to make the straight line back to the shore being the unknown but then we have even more unknowns so I have no idea what to do. Any guidance for me please :( Iv'e got the Exam tomorrow and this is the last part of the chapter and most of it is still unknown for me.

 

[Mark44]

Homework Statement

As ship is anchor off a long straight shoreline that runs north and south. From twi observation points. 15 miles apart on shore the bearings of the ship are N 31 ° E and S 53 ° E. What is the shortest distance from the ship to the shore.

Homework Equations

Sin θ Opp/ Hyp
Cos θ Adj/ Hyp
Tan θ Opp/ Adj

The Attempt at a Solution

I have drawn out the photo and it has a base of 15 miles and degree angles of 31, 53 and 96
I assumed the fastest route was a straight line back and that would cut the triangle into 2 different right triangles with their bases added together = 15 miles. I am confused if I should do a system of linear equations to make the straight line back to the shore being the unknown but then we have even more unknowns so I have no idea what to do.

This is how I would go, as well. You can divide up the 15 mile distance between the two stations as a and 15 - a. Call the distance from the shore to the ship d.
Write two equations in the two unknowns.
Solve.

Any guidance for me please :( Iv'e got the Exam tomorrow and this is the last part of the chapter and most of it is still unknown for me.

 

[verty]

Here is another way, find the rate that the distance parallel to the shore between the sides of the triangle reduces, then use that to find the distance. If this is confusing, you should ignore this suggestion...