1. **State the problem:** Simplify and analyze the expression $80\sin(\theta) + 20$. 2. **Formula and rules:** The sine function $\sin(\theta)$ oscillates between $-1$ and $1$. Multiplying by 80 scales this range to $[-80, 80]$. Adding 20 shifts the entire range up by 20. 3. **Intermediate work:** - Minimum value: $80 \times (-1) + 20 = -80 + 20 = -60$ - Maximum value: $80 \times 1 + 20 = 80 + 20 = 100$ 4. **Explanation:** The function $80\sin(\theta) + 20$ oscillates between $-60$ and $100$ as $\theta$ varies. The sine wave is scaled by 80 and shifted vertically by 20. 5. **Final answer:** The range of $80\sin(\theta) + 20$ is $$[-60, 100].$$