1. Stating the problem. Find the minimum value of $f(\theta)=(150\sin\theta+183.384)(253.8-150\cos\theta)$ and find the value(s) of $\theta$ where this minimum occurs. 2. Strategy and formula to use. We view $f(\theta)$ as a product $u(\theta)v(\theta)$ with $u(\theta)=150\sin\theta+183.384$ and $v(\theta)=253.8-150\cos\theta$. To find extrema we use the product rule and set the derivative to zero, i.e. $f'(\theta)=0$. 3. Compute the derivative using the product rule. Let $u(\theta)=150\sin\theta+183.384$ and $v(\theta)=253.8-150\cos\theta$. Then $u'(\theta)=150\cos\theta$ and $v'(\theta)=150\sin\theta$. By the product rule, $$f'(\theta)=u'(\theta)v(\theta)+u(\theta)v'(\theta)=150\big((150\sin\theta+183.384)\sin\theta+(253.8-150\cos\theta)\cos\theta\big).$$ 4. Critical-point condition simplification. Setting $f'(\theta)=0$ is equivalent to $$(150\sin\theta+183.384)\sin\theta+(253.8-150\cos\theta)\cos\theta=0.$$ Expand and simplify using $\sin^2\theta-\cos^2\theta=-\cos(2\theta)$ to obtain $$150\cos(2\theta)-183.384\sin\theta-253.8\cos\theta=0.$$ This is the transcendental equation we must solve for $\theta$. 5. Solve the equation numerically for the first critical point (choose principal root in $(-\pi,\pi]$). We solve $150\cos(2\theta)-183.384\sin\theta-253.8\cos\theta=0$ numerically (e.g., Newton or bisection). Bracketing and Newton iterations give a root at approximately $\theta\approx-1.41638\text{ rad}$. 6. Verify and evaluate $f$ at the critical point and compare neighbors to ensure a minimum. Compute $\sin(\theta)\approx-0.98836$ and $\cos(\theta)\approx0.15230$ at $\theta\approx-1.41638$. Then $u(\theta)=150\sin\theta+183.384\approx150(-0.98836)+183.384\approx35.130$. And $v(\theta)=253.8-150\cos\theta\approx253.8-150(0.15230)\approx230.955$. Therefore the function value is $$f(\theta)\approx35.130\times230.955\approx8113.46.$$ Checking nearby values (for example $\theta=-\tfrac{\pi}{2}$ and $\theta=0$) gives larger values, confirming this critical point is a local (and here global) minimum on one period. 7. Final answer. The function attains its minimum at approximately $\theta\approx-1.41638\text{ rad}$, and the minimum value is approximately $8113.46$.