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To solve for \(\theta\) in the given interval \([0^\circ, 360^\circ]\) based on the equation:

\[
\csc \theta = \frac{2 \sqrt{3}}{3}
\]

We first recognize that \(\csc \theta = \frac{1}{\sin \theta}\), so:

\[
\frac{1}{\sin \theta} = \frac{2 \sqrt{3}}{3}
\]

Now, solve for \(\sin \theta\):

\[
\sin \theta = \frac{3}{2 \sqrt{3}}
\]

Simplifying \(\sin \theta\):

\[
\sin \theta = \frac{\sqrt{3}}{2}
\]

Now, we need to find all values of \(\theta\) such that \(\sin \theta = \frac{\sqrt{3}}{2}\) in the interval \([0^\circ, 360^\circ]\).

The general angles where \(\sin \theta = \frac{\sqrt{3}}{2}\) are:

\[
\theta = 60^\circ \quad \text{(in the first quadrant)}
\]
and
\[
\theta = 180^\circ - 60^\circ = 120^\circ \quad \text{(in the second quadrant)}.
\]

Thus, the solutions for \(\theta\) are:

\[
\theta = 60^\circ \quad \text{and} \quad \theta = 120^\circ.
\]

Let me know if you want any further details or have questions! Here are five related questions:

1. What is the value of \(\sin(60^\circ)\) and why does it equal \(\frac{\sqrt{3}}{2}\)?
2. How do the reciprocal trigonometric functions like \(\csc\) relate to \(\sin\)?
3. Can we find other values of \(\theta\) if the interval changes to \([0^\circ, 720^\circ]\)?
4. What happens to the angle solutions if we work in radians instead of degrees?
5. How can the Pythagorean identity help verify the solutions for \(\sin \theta = \frac{\sqrt{3}}{2}\)?

**Tip:** Memorizing the values of trigonometric functions for common angles like \(30^\circ\), \(45^\circ\), and \(60^\circ\) is useful for solving many trigonometric equations efficiently.

Find all values of θ, if θ is in the interval [0°, 360°] and has the given function value. csc θ = (2√3)/3

Find all angles θ in the interval [0°, 360°] where the given function value csc(θ) = (2√3)/3 holds true. This problem covers core trigonometric concepts, including solving for sine and using reciprocal identities.

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