To solve the problem [tex]\(\cos^{-1}\left(\cos 43^{\circ}\right)\)[/tex], we need to understand the properties of the cosine function and its inverse.
1. Understanding the Range of Cosine and its Inverse:
- The cosine function, [tex]\(\cos(\theta)\)[/tex], is defined for all angles [tex]\(\theta\)[/tex].
- The inverse cosine function, [tex]\(\cos^{-1}(x)\)[/tex], also known as arccos, returns an angle [tex]\(\theta\)[/tex] such that [tex]\(0^\circ \leq \theta \leq 180^\circ\)[/tex].
2. Cosine is One-to-One in the Range of Arccos:
- The cosine function is one-to-one in the interval [tex]\([0^\circ, 180^\circ]\)[/tex].
- Therefore, for any angle [tex]\(\theta\)[/tex] in this interval, [tex]\(\cos^{-1}(\cos(\theta)) = \theta\)[/tex].
3. Given Angle within the Required Range:
- We are given the angle [tex]\(43^\circ\)[/tex].
- Since [tex]\(43^\circ\)[/tex] falls within the interval [tex]\([0^\circ, 180^\circ]\)[/tex], we can directly apply the property stated above.
4. Applying the Property:
- For [tex]\(\theta = 43^\circ\)[/tex], we use the relationship [tex]\(\cos^{-1}(\cos(\theta)) = \theta\)[/tex].
- Hence, [tex]\(\cos^{-1}(\cos(43^\circ)) = 43^\circ\)[/tex].
Therefore, the value is:
[tex]\[
\cos^{-1}\left(\cos 43^\circ\right) = 43^\circ
\][/tex]
