I have an oak tree in my yard I need to cut down; the ideal direction to fall it in is toward my workshop. The shortest distance measured from the base of the tree to the shop is 78 feet. I need to determine the height of the tree, I have a transit I can use to optically determine angles.
If I set up my transit 100 feet from the base of the tree, what would be the maximum angle in degrees that would tell me that the tree is too tall to fall in the direction desired
The ground from the tree to the shop is the base of a right triangle, the tree is the height of the right triangle, and the line of sight through your transit is the hypotenuse of the triangle. That puts the angle measured by your transit as the angle of the base and the hypotenuse. The adjacent side of that angle is the base and the opposite side is the Height of the tree.
The tangent of an angle in a right triangle is `(opposite)/(adjacent)` or in this case `(Height of the tree)/(distance to the transit)` If you place the transit 100 feet from the tree`` the tangent of the angle will be `x/100` and we only have 78 feet between the tree and the shop. So we don't want the tangent to exceed `78/100` We find `Arctan (78/100) ~~ 37.95` So to be safe, you don't want the transit to record an angle in excess of 37 degrees.