To find the exact values for the inverse cosecant, secant, and cotangent of \(\frac{\sqrt{3}}{3}\), we'll evaluate each one separately.
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Finding \(\csc^{-1}\left(\frac{\sqrt{3}}{3}\right)\):
The cosecant function is the reciprocal of the sine function. Therefore, we need to find the angle whose cosecant is \(\frac{\sqrt{3}}{3}\):
\[ \csc(\theta) = \frac{\sqrt{3}}{3} \implies \sin(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3} \]
This value is not possible, as the sine function only ranges from -1 to 1. Therefore, we conclude:
\[ \csc^{-1}\left(\frac{\sqrt{3}}{3}\right) \text{ does not have a defined value.} \]
Answer: The exact value cannot be determined.
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Finding \(\sec^{-1}\left(\frac{\sqrt{3}}{3}\right)\):
The secant function is the reciprocal of the cosine function. We are looking for the angle whose secant is \(\frac{\sqrt{3}}{3}\):
\[ \sec(\theta) = \frac{\sqrt{3}}{3} \implies \cos(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3} \]
This value is also not possible, as the cosine function also only ranges from -1 to 1. Therefore, we conclude:
\[ \sec^{-1}\left(\frac{\sqrt{3}}{3}\right) \text{ does not have a defined value.} \]
Answer: The exact value cannot be determined.
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Finding \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right)\):
The cotangent function is the reciprocal of the tangent function. We are looking for the angle whose cotangent is \(\frac{\sqrt{3}}{3}\):
\[ \cot(\theta) = \frac{\sqrt{3}}{3} \implies \tan(\theta) = \frac{3}{\sqrt{3}} = \sqrt{3} \]
The angle whose tangent is \(\sqrt{3}\) is \(\frac{\pi}{3}\):
\[ \cot^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{3} \]
Answer: \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right) = \frac{\pi}{3}\).
Summary of the Results:
- \(\csc^{-1}\left(\frac{\sqrt{3}}{3}\right)\): The exact value cannot be determined.
- \(\sec^{-1}\left(\frac{\sqrt{3}}{3}\right)\): The exact value cannot be determined.
- \(\cot^{-1}\left(\frac{\sqrt{3}}{3}\right)\): \(\frac{\pi}{3}\)