4. Find the exact value of \( \theta \) in the interval \( \left[\pi, \frac{3 \pi}{2}\right] \) and \( \csc \theta=-\frac{2 \sqrt{3}}{3} \). This is to be done without a calculator.

To find the exact value of \( \theta \) in the interval \( \left[\pi, \frac{3 \pi}{2}\right] \) such that \( \csc \theta = -\frac{2 \sqrt{3}}{3} \), we start by recalling the relationship between cosecant and sine: \[ \csc \theta = \frac{1}{\sin \theta} \] Thus, we can express the equation in terms of sine: \[ \sin \theta = \frac{1}{\csc \theta} = -\frac{3}{2 \sqrt{3}} \] To simplify \( -\frac{3}{2 \sqrt{3}} \), we can rationalize the denominator: \[ -\frac{3}{2 \sqrt{3}} = -\frac{3 \sqrt{3}}{2 \cdot 3} = -\frac{\sqrt{3}}{2} \] Now we have: \[ \sin \theta = -\frac{\sqrt{3}}{2} \] Next, we need to determine the angles where the sine function equals \( -\frac{\sqrt{3}}{2} \). The sine function is negative in the third and fourth quadrants. The reference angle for \( \sin \theta = \frac{\sqrt{3}}{2} \) is \( \frac{\pi}{3} \). In the third quadrant, the angle corresponding to this reference angle is: \[ \theta = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \] In the fourth quadrant, the angle would be: \[ \theta = 2\pi - \frac{\pi}{3} = \frac{6\pi}{3} - \frac{\pi}{3} = \frac{5\pi}{3} \] However, since we are only interested in the interval \( \left[\pi, \frac{3 \pi}{2}\right] \), we only consider \( \frac{4\pi}{3} \). Thus, the exact value of \( \theta \) is: \[ \boxed{\frac{4\pi}{3}} \]