1. **Problem statement:** Given a point $P(0) = \left(-\frac{1}{2}, -\frac{1}{\sqrt{2}}\right)$ on the unit circle corresponding to angle $\theta$, find the coordinates of $P(\theta + \frac{\pi}{2})$ and $P(\theta - \frac{\pi}{2})$. 2. **Recall the unit circle properties:** For any angle $\alpha$, the coordinates of the point on the unit circle are $P(\alpha) = (\cos \alpha, \sin \alpha)$. 3. **Use angle addition formulas:** - $\cos(\theta + \frac{\pi}{2}) = -\sin \theta$ - $\sin(\theta + \frac{\pi}{2}) = \cos \theta$ - $\cos(\theta - \frac{\pi}{2}) = \sin \theta$ - $\sin(\theta - \frac{\pi}{2}) = -\cos \theta$ 4. **Find $\cos \theta$ and $\sin \theta$ from $P(0)$:** Given $P(0) = (\cos \theta, \sin \theta) = \left(-\frac{1}{2}, -\frac{1}{\sqrt{2}}\right)$, so $\cos \theta = -\frac{1}{2}$ and $\sin \theta = -\frac{1}{\sqrt{2}}$. 5. **Calculate $P(\theta + \frac{\pi}{2})$:** $$ \cos\left(\theta + \frac{\pi}{2}\right) = -\sin \theta = -\left(-\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} $$ $$ \sin\left(\theta + \frac{\pi}{2}\right) = \cos \theta = -\frac{1}{2} $$ Thus, $$ P\left(\theta + \frac{\pi}{2}\right) = \left(\frac{1}{\sqrt{2}}, -\frac{1}{2}\right) $$ 6. **Calculate $P(\theta - \frac{\pi}{2})$:** $$ \cos\left(\theta - \frac{\pi}{2}\right) = \sin \theta = -\frac{1}{\sqrt{2}} $$ $$ \sin\left(\theta - \frac{\pi}{2}\right) = -\cos \theta = -\left(-\frac{1}{2}\right) = \frac{1}{2} $$ Thus, $$ P\left(\theta - \frac{\pi}{2}\right) = \left(-\frac{1}{\sqrt{2}}, \frac{1}{2}\right) $$ **Final answers:** - $P\left(\theta + \frac{\pi}{2}\right) = \left(\frac{1}{\sqrt{2}}, -\frac{1}{2}\right)$ - $P\left(\theta - \frac{\pi}{2}\right) = \left(-\frac{1}{\sqrt{2}}, \frac{1}{2}\right)$