IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.
Sagot :
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]
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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.