1. **Problem statement:** Given $\cos \theta = -\frac{2}{5}$ and $\sin \theta > 0$, find the quadrant where $\theta$ lies, then find $\sin \theta$, $\tan \theta$, $\sec \theta$, $\csc \theta$, and $\cot \theta$ in decimal form. 2. **Determine the quadrant:** - $\cos \theta = -\frac{2}{5}$ is negative. - $\sin \theta > 0$ is positive. - Cosine is negative and sine is positive in **Quadrant II**. 3. **Find $\sin \theta$ using Pythagorean identity:** $$\sin^2 \theta + \cos^2 \theta = 1$$ $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{2}{5}\right)^2 = 1 - \frac{4}{25} = \frac{21}{25}$$ Since $\sin \theta > 0$, $$\sin \theta = \sqrt{\frac{21}{25}} = \frac{\sqrt{21}}{5} \approx 0.9165$$ 4. **Find $\tan \theta$:** $$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{\sqrt{21}}{5}}{-\frac{2}{5}} = -\frac{\sqrt{21}}{2} \approx -2.2913$$ 5. **Find $\sec \theta$:** $$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{2}{5}} = -\frac{5}{2} = -2.5$$ 6. **Find $\csc \theta$:** $$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{21}}{5}} = \frac{5}{\sqrt{21}} = \frac{5\sqrt{21}}{21} \approx 1.0911$$ 7. **Find $\cot \theta$:** $$\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-\frac{\sqrt{21}}{2}} = -\frac{2}{\sqrt{21}} = -\frac{2\sqrt{21}}{21} \approx -0.4364$$ **Final answers:** - Quadrant: II - $\sin \theta \approx 0.9165$ - $\tan \theta \approx -2.2913$ - $\sec \theta = -2.5$ - $\csc \theta \approx 1.0911$ - $\cot \theta \approx -0.4364$