To find the measure of the angle [tex]$\theta$[/tex] at which the surveyor stands using the given cosine values, we need to use the given inverse cosine (arccos) values. The arccos function helps us find an angle when we know its cosine value.
Here are the provided cosine values and their corresponding angles:
- [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]
By looking at these relationships, we can conclude the following:
1. When the cosine value is [tex]\(0.75\)[/tex], the angle is [tex]\(41^\circ\)[/tex].
2. When the cosine value is [tex]\(0.125\)[/tex], the angle is [tex]\(83^\circ\)[/tex].
3. When the cosine value is [tex]\(0.563\)[/tex], the angle is [tex]\(56^\circ\)[/tex].
4. When the cosine value is [tex]\(0.15\)[/tex], the angle is [tex]\(89^\circ\)[/tex].
Thus, the measure of the angle at which the surveyor stands would be one of these values depending on the cosine value observed:
[tex]\[
41^\circ, 83^\circ, 56^\circ, \text{or } 89^\circ
\][/tex]
These angles are approximations to the nearest degree based on the given cosine values and their respective inverse cosine functions. Hence, the surveyor could be standing at any of these angles depending on the measured cosine value.
