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A boat going parallel to shore spots a lighthouse ahead on shore. The angle of the line from lighthouse to boat is 30 degrees. The boat sails 3mi, and now angle is 90. How far offshore is boat?

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Draw it out first. Can you see how the boat's path in the problem forms a triangle? – Drew Christianson Sep 21 '12 at 20:38
30 degrees between the line and the shore, presumably? – rschwieb Sep 21 '12 at 20:38

Consider the right angle triangle seen here:

Let, B: the location of the lighthouse, A: your past location (when the line formed an angle of 30 deg), C: your current location at 90 deg.

You know the angle at vertex A, ang(A) = 30 deg, and, b = 3mi, you are asked to determine small a. What is the relationship between ang(A), a, and b?

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Here’s a diagram of the setting:

enter image description here

You want the distance $x$. It’s the length of one of the legs of a nice right triangle, and you know the length of the other leg. If you recognize this as a $30$-$60$-$90$ triangle and know the proportions of the sides in such a triangle $-$ and this is useful information that you probably should learn $-$ then you can get $x$ immediately. If not, you’ll need to use one of the trig functions of $30$°; do you know which one is useful here?

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$\sqrt{3}$ miles. A picture helps.

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$\tan(\pi/6) = x/3$; $x = 3 \tan( \pi/6) = 3/\sqrt 3 = \sqrt 3$

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