Suppose a surveyor as 50 feet from the base of a tree, and the angle to the top is calculated to be $51.8^{\circ}$. How accurately must the angle be measured if the height error is less than 6%? The answer must be given in $d\theta$ radians.
Let the adjacent side= 50, the opposing side be x, and the ground angle be $51.8^{\circ}$. Therefore x is calculated as $\frac{x}{50}= \tan{(51.8)}$ $x= 50~ \tan{(51.8)}$ $x= 63.5386$
Since there is a +/- 6% error allowance, $x+0.06= 63.5386+3.812=~67.3506\text{ feet for the upper bound}$ $x-0.06= 63.5386-3.812=~59.7266\text{ feet for the lower bound}$
Using the inverse tangent function to determine the new angles necessary,
$tan^{-1}\frac{67.3506}{50}= 53.410^{\circ} \text{ which is an increase of } 0.923 \text{ degrees}$ $tan^{-1}\frac{59.7266}{50}= 50.066^{\circ} \text{ which is an decrese of } 0.874 \text{ degrees}$
As I have two different bounds, which is the the one to be used for the solution?