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.

To find the value of all six trigonometric functions for the angle \( \theta \) where \( 180^{\circ} < \theta < 360^{\circ} \) and \( \cos \theta = -\frac{2}{5} \), we will firstly determine the values of sine and tangent using the Pythagorean identity. 1. **Finding \(\sin \theta\)**: We know from the Pythagorean identity that: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting \(\cos \theta\): \[ \sin^2 \theta + \left(-\frac{2}{5}\right)^2 = 1 \] \[ \sin^2 \theta + \frac{4}{25} = 1 \] \[ \sin^2 \theta = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25} \] Therefore, taking the square root: \[ \sin \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5} \] Since \( \theta \) is in the third quadrant (where sine is negative), we have: \[ \sin \theta = -\frac{\sqrt{21}}{5} \] 2. **Finding \(\tan \theta\)**: Now, we can calculate \(\tan \theta\) using the sine and cosine values: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = \frac{\sqrt{21}}{2} \] Now we can summarize the known values: - \(\cos \theta = -\frac{2}{5}\) - \(\sin \theta = -\frac{\sqrt{21}}{5}\) - \(\tan \theta = \frac{\sqrt{21}}{2}\) 3. **Finding the remaining trigonometric functions**: - **Cosecant**: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{21}}{5}} = -\frac{5}{\sqrt{21}} = -\frac{5\sqrt{21}}{21} \] - **Secant**: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{2}{5}} = -\frac{5}{2} \] - **Cotangent**: \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{\sqrt{21}}{2}} = \frac{2}{\sqrt{21}} = \frac{2\sqrt{21}}{21} \] Finally, we summarize the results: \[ \sin \theta = -\frac{\sqrt{21}}{5}, \quad \cos \theta = -\frac{2}{5}, \quad \tan \theta = \frac{\sqrt{21}}{2} \] \[ \csc \theta = -\frac{5\sqrt{21}}{21}, \quad \sec \theta = -\frac{5}{2}, \quad \cot \theta = \frac{2\sqrt{21}}{21} \] Thus, the values of all six trigonometric functions for the angle \( \theta \) are: - \(\sin \theta = -\frac{\sqrt{21}}{5}\) - \(\cos \theta = -\frac{2}{5}\) - \(\tan \theta = \frac{\sqrt{21}}{2}\) - \(\csc \theta = -\frac{5\sqrt{21}}{21}\) - \(\sec \theta = -\frac{5}{2}\) - \(\cot \theta = \frac{2\sqrt{21}}{21}\)