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We are given that \(\csc \theta = \frac{2\sqrt{3}}{3}\) and need to find \(\sec \theta\). Let's solve this step by step.

1. **Relationship between csc and sin:**
   Since \(\csc \theta = \frac{1}{\sin \theta}\), we can express \(\sin \theta\) as:
   \[
   \sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{2\sqrt{3}}{3}} = \frac{3}{2\sqrt{3}}.
   \]
   To simplify, multiply numerator and denominator by \(\sqrt{3}\):
   \[
   \sin \theta = \frac{3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}.
   \]

2. **Finding cosθ using the unit circle identity:**
   From the Pythagorean identity:
   \[
   \sin^2 \theta + \cos^2 \theta = 1,
   \]
   substitute \(\sin \theta = \frac{\sqrt{3}}{2}\):
   \[
   \left(\frac{\sqrt{3}}{2}\right)^2 + \cos^2 \theta = 1,
   \]
   \[
   \frac{3}{4} + \cos^2 \theta = 1,
   \]
   \[
   \cos^2 \theta = 1 - \frac{3}{4} = \frac{1}{4},
   \]
   \[
   \cos \theta = \pm \frac{1}{2}.
   \]

3. **Finding secθ:**
   Since \(\sec \theta = \frac{1}{\cos \theta}\), there are two possible values for \(\sec \theta\):
   \[
   \sec \theta = \pm 2.
   \]
   The correct answer would depend on the quadrant of \(\theta\), but typically when working with the unit circle, positive values are taken when no specific quadrant is given. Hence, we take the positive value:
   \[
   \sec \theta = +2.
   \]

Therefore, the answer is **C. +2**.

Would you like more details or have any questions?

Here are five related questions for further practice:

1. How do you determine the sign of cosine based on the quadrant?
2. What is the relationship between \(\sec \theta\) and \(\cos \theta\)?
3. Can you find \(\tan \theta\) given the value of \(\sin \theta\)?
4. How does \(\csc \theta\) change as \(\theta\) approaches \(\frac{\pi}{2}\)?
5. What are the key trigonometric identities involving secant and cosecant?

**Tip**: Always remember the quadrant rules when working with trigonometric functions, as they determine the sign of your answer!

In a unit circle, if csc θ = (2√3)/3, then sec θ = ____.

This problem focuses on calculating the secant of an angle when given its cosecant value in a unit circle. It involves key trigonometric concepts such as the Pythagorean identity and unit circle relationships. Ideal for students in Grades 10-12.

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