TRIG Question!

Dec 2011
12
0
Please help.

Here is the problem.

1. Ted is flying a kite with a string that makes an angle of 40 degrees with the horizontal. If the kite is 750 feet high, how long is the string?

and
2. Rita is estimating the height of a tree. From a point due east of the tree, the angle of elevation to the top is 27degrees. She walks 200 feet toward the tree and the angle of elevation increases to 39 degrees.
How tall is tree?

how far from the base of the tree was rita standing when she first observed the tree?

How do I know when to use Cosine and Tangent? I don't get it - and how do I know when to divide/multiply them? Please help.
 
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greg1313

Forum Staff
Oct 2008
8,008
1,174
London, Ontario, Canada - The Forest City
1. Ted is flying a kite with a string that makes an angle of 40 degrees with the horizontal. If the kite is 750 feet high, how long is the string?

Let x be the length of the string. Using the sine ratio, sin(40) = 750/x, x = 750/sin(40) ≈ 1166.7929.

2. Rita is estimating the height of a tree. From a point due east of the tree, the angle of elevation to the top is 27degrees. She walks 200 feet toward the tree and the angle of elevation increases to 39 degrees.
How tall is tree?

Here's one way to do it:

Let T be the height of the tree. Let x be the distance from Rita to the tree after she walks 200 feet towards the tree.

tan(39) = T/x, x = T/tan(39) (*)

tan(27) = T/(200 + x) (**)

Substitute x from (*) into (**):

tan(27) = T/(200 + T/tan(39))

(200 + T/tan(39))tan(27) = T

200tan(27) + tan(27)T/tan(39) = T

200tan(27) = T - tan(27)T/tan(39)

200tan(27) = T(1 - tan(27)/tan(39))

T = 200tan(27)/(1 - tan(27)/tan(39)) ≈ 274.8335.
 
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skipjack

Forum Staff
Dec 2006
21,394
2,413
2. Make a diagram if you find it awkward to imagine one.
As tan is opposite/adjacent,
Rita's initial distance from tree = T*tan(90°-27°) = T*tan(63°), and
Rita's closer distance from tree = T*tan(90°-39°) = T*tan(51°).

As these differ by 200 ft, T = 200 ft /(tan(63°) - tan(51°)) = 274.8334908 ft approximately.

The rest is easy. The results should be rounded appropriately.