Trig Tan Math problem

tan 82° = x / 4250

(tan 82°) * 4250 = x

x ≈ 30240.32 ft. ( I don't know if I interpreted the problem right.)

Second problem:

This is a right triangle problem.

In this case the base of triangle is 12 feet and the hypotenuse is 13 feet.

use pythagorean theorem:  a² + b² = c²

12² + b² = 13²

144 + b² = 169

b² = 169 - 144

b² = 25

b = 5 feet.

so, the tree is 5 feet tall.

                  B
                 /|      
               /  |      
             /    |      ↑    
           /      |   4250 ft
         /        |      ↓
    A / 82⁰    | C                 

  BC / AC = tan 82⁰ ---> AC = BC / tan 82⁰
B                                  AC ≈ 4250 ÷ 7.11537 = 597.3 ft
|\         
|   \       
|      \    13 ft
|          \   
|    12 ft   \  A  
C

 a² + b² = c² ---> a² = 13² - 12² = 25 ---> a = 5 ft
OR
cos A = AB / AC
cos A = 12 / 13 = 0.923077
measure of angel A = 22.62⁰ 
sin 22.62⁰ = BC / AB ---> BC = AB * sin 22.62⁰ ---> BC = 13 * 0.3846 = 5 ft

You have two problems here.  For the first one, a very good explanation can be found here on  how to proceed

http://www.mathsisfun.com/algebra/trig-finding-side-right-triangle.html

You have the angle (82 degrees) and the adjacent side (4250 feet) but need the opposite side's measurement, so you are working with the tangent.

The second is classic Pythagorus: tree height is A  (? feet), shadow cast  (12 feet) is B, shadow tip to tree top is C (13 feet or the hypotenuse{H}).

Draw the picture, it will help makes things clearer.

So, A-squared plus B-squared equals C-squared:

A² + B² = C²

A² + 12² = 13²

A² + 144 = 169

Move B to the right side of the equation

A² = 169 - 144

Subtract B from C

A² = 25

Take the square root of both sides of the equation

A = 5

The tree is 5 feet tall.