Consider angle \( \theta \), where \( 180^{\circ}<\theta<360^{\circ} \) and \( \cos \theta=-\frac{2}{5} \). Find the value of all six trigonometric functions for the angle \( \theta \). Simplify radicals and rationalize denominators, if necessary. Give exact solutions (no rounded decimals).

To find the six trigonometric functions for the angle \( \theta \) where \( 180^{\circ}<\theta<360^{\circ} \) and \( \cos \theta=-\frac{2}{5} \), we start by using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1. \] Since \( \cos \theta = -\frac{2}{5} \), we first compute \( \cos^2 \theta \): \[ \cos^2 \theta = \left(-\frac{2}{5}\right)^2 = \frac{4}{25}. \] We can now find \( \sin^2 \theta \): \[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}. \] Taking the square root to determine \( \sin \theta \), we have: \[ \sin \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}. \] Since \( \theta \) is in the third quadrant (where sine is negative), we get: \[ \sin \theta = -\frac{\sqrt{21}}{5}. \] Now that we have \( \sin \theta \) and \( \cos \theta \), we can find the remaining trigonometric functions: 1. **Tangent**: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = \frac{\sqrt{21}}{2}. \] 2. **Cosecant** (reciprocal of sine): \[ \csc \theta = \frac{1}{\sin \theta} = -\frac{5}{\sqrt{21}} \implies \csc \theta = -\frac{5\sqrt{21}}{21}. \] 3. **Secant** (reciprocal of cosine): \[ \sec \theta = \frac{1}{\cos \theta} = -\frac{5}{2}. \] 4. **Cotangent** (reciprocal of tangent): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{2}{\sqrt{21}} \implies \cot \theta = \frac{2\sqrt{21}}{21}. \] Now, summarizing all six trigonometric functions: - \( \sin \theta = -\frac{\sqrt{21}}{5} \) - \( \cos \theta = -\frac{2}{5} \) - \( \tan \theta = \frac{\sqrt{21}}{2} \) - \( \csc \theta = -\frac{5\sqrt{21}}{21} \) - \( \sec \theta = -\frac{5}{2} \) - \( \cot \theta = \frac{2\sqrt{21}}{21} \) Thus, the exact values of the trigonometric functions for the angle \( \theta \) are: \[ \boxed{\sin \theta = -\frac{\sqrt{21}}{5}, \quad \cos \theta = -\frac{2}{5}, \quad \tan \theta = \frac{\sqrt{21}}{2}, \quad \csc \theta = -\frac{5\sqrt{21}}{21}, \quad \sec \theta = -\frac{5}{2}, \quad \cot \theta = \frac{2\sqrt{21}}{21}.} \]