1. If \( \cos 2 \theta=-\frac{5}{6} \), where \( 2 \theta \epsilon\left[180^{\circ} ; 360^{\circ}\right] \), calculate, without using a calculat values in the simplest form of:

To find \( \theta \) given that \( \cos 2\theta = -\frac{5}{6} \) and \( 2\theta \) is in the interval \([180^\circ, 360^\circ]\), we first determine \( 2\theta \). The cosine function is negative in the third and fourth quadrants, which aligns with our interval. To find \( 2\theta \), we can use the inverse cosine function. However, \(\cos^{-1}\) could yield angles outside our required range, so we proceed by writing: \[ 2\theta = 360^\circ - \cos^{-1}\left(-\frac{5}{6}\right) \] and \[ 2\theta = 180^\circ + \cos^{-1}\left(-\frac{5}{6}\right). \] Using properties of cosine, we choose the angle in the second quadrant, calculating: 1. \( 2\theta_1 = 360^\circ - x \) (angle in fourth quadrant), 2. \( 2\theta_2 = 180^\circ + x \) (angle in third quadrant). Calculating \( \theta \) involves halving our results. If we denote \( x = \cos^{-1}\left(-\frac{5}{6}\right) \) as some angle, then the final expressions become \( \theta = 90^\circ - \frac{x}{2} \) for the quadrant in which cosine is positive. Thus, the solutions for \( 2\theta \) yield two possibilities, halving to find \( \theta \) gives us potential values we'd calculate to find a more exact number form \( \left( n \, \text{with} \, x \text{ dependent on calculative evaluations} \right) \), leading us to: \[ \theta_1 = 360^\circ - \frac{x}{2} \text{ or } \theta_2 = \frac{180^\circ + x}{2}. \] Resulting values can be computed for accurate degrees or radians, keeping in the simplest terms expected for final \( \theta \).