To solve for [tex]\(\cos^{-1}(1)\)[/tex], we need to determine the angle whose cosine is 1.
1. Understanding the problem: The inverse cosine function, denoted as [tex]\(\cos^{-1}\)[/tex], gives us the angle whose cosine is a specific value. In this case, we need the angle [tex]\( \theta \)[/tex] such that [tex]\(\cos(\theta) = 1\)[/tex].
2. Recognizing the cosine value of 1: We know from the properties of the cosine function that [tex]\(\cos(0) = 1\)[/tex]. This is because on the unit circle, the angle of 0 radians points directly to the right along the positive x-axis, which corresponds to the cosine value of 1.
3. Identifying the relevant angle: Since we seek the principal value of the inverse cosine function and know that [tex]\(\cos(0) = 1\)[/tex], we conclude that [tex]\(\cos^{-1}(1)\)[/tex] is 0 radians. The range of the inverse cosine function is [tex]\([0, \pi]\)[/tex] radians, and 0 radians falls within this range.
So, the value of [tex]\(\cos^{-1}(1)\)[/tex] is:
[tex]\[
0 \, \text{radians}
\][/tex]
