# Trigonometry question?

How would I solve the following trig questions.

1. A tree broken over by the winds forms a right angle with the ground.If the broken part makes angle of 50 degrees with the ground and if the top of the tree is now 20 feet from the base how tall was the tree?

2. Two straight roads intersect to form an angle 0f 75 degrees. Find the shortest distance from one road to a gas station on the other road 1000 m from the junction.

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$1.$ Draw a picture. Let $b$ be the current (broken) height of the tree, that is, the distance from the ground to the break, which is really only a sharp bend. Then $$\frac{b}{20}=\tan(50^\circ).$$ Let $\ell$ be the length of the "leaning" part. Then $$\frac{20}{\ell}=\cos(50^\circ).$$ Finally, the original height of the tree is $b+\ell$.

The calculation: We have $b=20\tan(50^\circ)\approx 23.835$, and $\ell=\frac{20}{\cos(50^\circ)}\approx 31.114$, giving a sum of $\approx 54.95$.

$2.$ Draw a picture. Let $O$ be the point of intersection of the streets, $G$ the gas station, and $P$ the point on the other road nearest to the gas station. Then $\angle OPG$ is a right angle. Thus $$\frac{PG}{OG}=\sin(75^\circ).$$

The calculator gives the distance as approximately $965.9$. Presumably you have instructions about what kind of rounding to do.

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I am still slight confused as how you would get the answer. –  Lucas Jun 23 '12 at 19:15
To which question? –  André Nicolas Jun 23 '12 at 19:17
the second I see tan 75 but the final answer in my book is 970 meter I am not sure how to get there. –  Lucas Jun 23 '12 at 19:19
also with the first one the final answer in my book is 56 feet but I cant make the numbers match –  Lucas Jun 23 '12 at 19:20
@Lucas: Question 2 answered, I had mixed up two letters. For the first question, definitely don't get 56, get almost $55$. –  André Nicolas Jun 23 '12 at 19:37