1. **Problem Statement:** Solve the equation $\cos\theta = \frac{\theta}{2}$ for $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ graphically and find the approximate solutions. 2. **Formula and Explanation:** We want to find values of $\theta$ where the curve $y = \cos\theta$ intersects the line $y = \frac{\theta}{2}$. The cosine function oscillates between -1 and 1, and the line $\frac{\theta}{2}$ is linear with slope $\frac{1}{2}$. 3. **Domain:** $\theta$ is restricted to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \approx [-1.5708, 1.5708]$. 4. **Intermediate Work:** - At $\theta = 0$, $\cos 0 = 1$ and $\frac{0}{2} = 0$, so $\cos\theta > \frac{\theta}{2}$. - At $\theta = \frac{\pi}{2} \approx 1.5708$, $\cos\frac{\pi}{2} = 0$ and $\frac{1.5708}{2} \approx 0.7854$, so $\cos\theta < \frac{\theta}{2}$. - At $\theta = -\frac{\pi}{2} \approx -1.5708$, $\cos\left(-\frac{\pi}{2}\right) = 0$ and $\frac{-1.5708}{2} = -0.7854$, so $\cos\theta > \frac{\theta}{2}$. 5. **Finding Intersection Points:** - By checking values and using graphical intuition, the curves intersect near $\theta \approx 0.9$ and $\theta \approx -1.2$. 6. **Final Answer:** The approximate solutions to $\cos\theta = \frac{\theta}{2}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ are: $$\theta \approx -1.2, \quad \theta \approx 0.9$$