1. Problem: Given point $P(-2,2)$ lies on the terminal arm of angle $\theta$ in standard position, determine which statement about $\tan\theta$, $\sin\theta$, $\cos\theta$, and $\csc\theta$ is correct. 2. Formula and rules: - $\tan\theta = \frac{y}{x}$ - $\sin\theta = \frac{y}{r}$ - $\cos\theta = \frac{x}{r}$ - $\csc\theta = \frac{r}{y}$ where $r = \sqrt{x^2 + y^2}$ is the distance from the origin to point $P$. 3. Calculate $r$: $$r = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}$$ 4. Calculate each ratio: - $\tan\theta = \frac{2}{-2} = -1$ - $\sin\theta = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ after rationalizing denominator - $\cos\theta = \frac{-2}{2\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}$ after rationalizing denominator - $\csc\theta = \frac{2\sqrt{2}}{2} = \sqrt{2}$ 5. Check each option: - a) $\tan\theta = 1$ is false because $\tan\theta = -1$ - b) $\sin\theta = -\frac{2}{\sqrt{8}} = -\frac{2}{2\sqrt{2}} = -\frac{1}{\sqrt{2}}$ is false because $\sin\theta$ is positive - c) $\cos\theta = -2$ is false because $\cos\theta = -\frac{\sqrt{2}}{2}$ - d) $\csc\theta = \frac{\sqrt{8}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}$ is true Answer: d) $\csc\theta = \frac{\sqrt{8}}{2}$ q_count: 1