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?
Best Answer
If you want to be sure you are within $6\%$ and want symmetric bounds, you have to use the smaller bound. The difference is basically due to the second derivative of the tangent function-if the second derivative were zero the allowable error would be symmetric. You could use asymmetric bounds and say the measured angle must be $50.066^{\circ}$ to $53.410^{\circ}$, or $51.8^{\circ}+.923-.874$. I think most people would just round down and say $51.8^{\circ}\pm 0.8^{\circ}$, as your angle is just given to one place.