To determine the measure of the angle of the triangular plot where the surveyor stands, we can use the trigonometric function cosine and its inverse, specifically the arccosine function, which gives us the angle whose cosine is a given number. The provided information gives us several cosine values along with their corresponding angles.
The provided values are:
- [tex]\(\cos^{-1}(0.75) = 41^\circ\)[/tex]
- [tex]\(\cos^{-1}(0.125) = 83^\circ\)[/tex]
- [tex]\(\cos^{-1}(0.563) = 56^\circ\)[/tex]
- [tex]\(\cos^{-1}(0.15) = 89^\circ\)[/tex]
We are asked to approximate the measure of the angle at which the surveyor stands.
Comparing the values given above:
- [tex]\( \cos(41^\circ) = 0.75 \)[/tex]
- [tex]\( \cos(83^\circ) = 0.125 \)[/tex]
- [tex]\( \cos(56^\circ) = 0.563 \)[/tex]
- [tex]\( \cos(89^\circ) = 0.15 \)[/tex]
Among these options, the angle we are interested in is one of the four provided measurements, and our task is to determine which one of these precise angles is the correct measure for the surveyor's angle.
Given that the correct measure of the angle at which the surveyor stands is the one corresponding to [tex]\(\cos^{-1}(0.15)\)[/tex], we see that the correct angle is:
[tex]\[ \cos^{-1}(0.15) = 89^\circ \][/tex]
Thus, the measure of the angle at which the surveyor stands is:
[tex]\[ \boxed{89^\circ} \][/tex]
