1. **State the problem:** Find the derivative of the function $$f(\theta) = 80\sin(\theta) + 20$$ with respect to $$\theta$$. 2. **Recall the derivative rules:** - The derivative of $$\sin(\theta)$$ with respect to $$\theta$$ is $$\cos(\theta)$$. - The derivative of a constant is 0. - The derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. 3. **Apply the derivative:** $$\frac{d}{d\theta} \left(80\sin(\theta) + 20\right) = 80 \frac{d}{d\theta} \sin(\theta) + \frac{d}{d\theta} 20$$ 4. **Calculate each term:** $$= 80 \cos(\theta) + 0$$ 5. **Final answer:** $$\boxed{80 \cos(\theta)}$$ This means the rate of change of the function $$80\sin(\theta) + 20$$ with respect to $$\theta$$ is $$80 \cos(\theta)$$.