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To find the exact values of the remaining trigonometric functions given that \(\sin \theta = \frac{\sqrt{2}}{6}\) and \(\cos \theta > 0\), we proceed as follows:

### 1. Find \(\cos \theta\):
Using the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Substitute \(\sin \theta = \frac{\sqrt{2}}{6}\):
\[
\left( \frac{\sqrt{2}}{6} \right)^2 + \cos^2 \theta = 1
\]
\[
\frac{2}{36} + \cos^2 \theta = 1
\]
\[
\frac{1}{18} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \frac{1}{18} = \frac{17}{18}
\]
Since \(\cos \theta > 0\):
\[
\cos \theta = \sqrt{\frac{17}{18}} = \frac{\sqrt{17}}{3\sqrt{2}} = \frac{\sqrt{34}}{6}
\]

### 2. Find \(\tan \theta\):
\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{2}}{6}}{\frac{\sqrt{34}}{6}} = \frac{\sqrt{2}}{\sqrt{34}} = \frac{\sqrt{2}}{\sqrt{34}} \cdot \frac{\sqrt{34}}{\sqrt{34}} = \frac{\sqrt{68}}{34} = \frac{2\sqrt{17}}{34} = \frac{\sqrt{17}}{17}
\]

### 3. Find \(\csc \theta\):
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{6}{\sqrt{2}} = 3\sqrt{2}
\]

### 4. Find \(\sec \theta\):
\[
\sec \theta = \frac{1}{\cos \theta} = \frac{6}{\sqrt{34}} = \frac{6\sqrt{34}}{34} = \frac{3\sqrt{34}}{17}
\]

### 5. Find \(\cot \theta\):
\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{\sqrt{17}}{17}} = \frac{17}{\sqrt{17}} = \sqrt{17}
\]

Would you like any further details on these calculations, or do you have any specific questions?

### Related Questions:
1. How would the values change if \(\cos \theta < 0\)?
2. Can you verify these trigonometric identities using a unit circle approach?
3. What are the values of the trigonometric functions in other quadrants?
4. How do these calculations change for other trigonometric identities, such as \(\cos 2\theta\)?
5. Can you show the geometric interpretation of these values on a coordinate plane?

**Tip:** For trigonometric functions, always use identities like the Pythagorean identity and reciprocal identities to simplify your calculations and ensure the values are correct.

Find the exact value of each of the remaining trigonometric functions of θ. Rationalize denominators when applicable.

Learn how to calculate the exact values of trigonometric functions for a given sin(θ), where sin(θ) = sqrt(2)/6, using the Pythagorean identity and rationalizing denominators. This comprehensive guide is ideal for students in Grades 10-12, covering core trigonometric concepts and operations.

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