# Finding the height given the angle of elevation and depression.

Please, I need help for this problem. I'm a little confused about it :(

From a point A 10ft. above the water the angle of elevation of the top of a lighthouse is 46 degrees and the angle of depression of its image is 50 degrees. Find the height of the lighthouse and its horizontal distance from the observer.

I don't know where to start, because the problem doesn't have opposite, hypotenuse, or adjacent side written on it, and I think I cannot use TOA since there were no "Adj" or "Opp" side on the problem.

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Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people are much more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many would consider your post rude because it is a command ("Find..."), not a request for help, so please consider rewriting it. – Zev Chonoles♦ Jul 9 '12 at 8:35
It looks like you can set up two equations in two unknowns here. Let $x$ be the lighthouse's height and $y$ be the distance to the lighthouse. Then if the picture in my head is right, $x-10$ and $y$ are the opposite and adjacent sides to a $46^\circ$ angle, and $x+10$ and $y$ are the opp. and adj. sides to a $50^\circ$ angle. – Eugene Shvarts Jul 9 '12 at 8:36

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@jhong: You should not repost your question if you have something to add - there is an "edit" button on the bottom left of the question, underneath the "homework" and "trigonometry" tags. I have taken all of the changes you made in your reposted question and put them here, and closed your reposted question as a duplicate. – Zev Chonoles♦ Jul 9 '12 at 10:31

## 2 Answers

Let $h$ be the height, $d$ the horizontal distance. Then you arrive at two equations (why?): $$d\tan 46^o=h-10$$$$d\tan 50^o=h+10$$.

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Saurabh Hota's diagram describes it perfectly! – Aneesh Karthik C Jul 9 '12 at 9:16

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Write EG in terms of angle $50^\circ$.Find $x$. Then the height will be $x\tan 46^\circ + 10$
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