\( 8.5 \sin \left(\theta-20^{\circ}\right)=\cos 3 \theta \). Find the general solution.

To solve the equation \( 8.5 \sin(\theta - 20^{\circ}) = \cos(3\theta) \), we will follow these steps: 1. **Rewrite the equation**: We can express the equation as: \[ 8.5 \sin(\theta - 20^{\circ}) - \cos(3\theta) = 0 \] 2. **Use trigonometric identities**: We can use the identity for \(\cos(3\theta)\): \[ \cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta) \] However, this might complicate the equation. Instead, we can look for solutions directly. 3. **Set up the general solution**: The equation involves both sine and cosine functions, which suggests that we can find solutions by considering the periodic nature of these functions. 4. **Finding specific solutions**: We can find specific angles that satisfy the equation. 5. **General solution**: The general solution for trigonometric equations typically involves adding integer multiples of the period of the sine and cosine functions. Let's analyze the equation further to find specific solutions. ### Step 1: Find specific solutions We can try to find values of \(\theta\) that satisfy the equation. 1. **Assume a value for \(\theta\)**: Let's start with \(\theta = 20^{\circ}\): \[ 8.5 \sin(20^{\circ} - 20^{\circ}) = \cos(3 \times 20^{\circ}) \implies 8.5 \cdot 0 = \cos(60^{\circ}) \implies 0 = 0 \] This is a valid solution. 2. **Next, consider the periodic nature**: The sine function has a period of \(360^{\circ}\) and the cosine function has a period of \(360^{\circ}\) as well. Therefore, we can express the general solution as: \[ \theta = 20^{\circ} + k \cdot 360^{\circ} \quad \text{for any integer } k \] ### Step 2: Find other solutions To find other solutions, we can also consider the angles where the sine and cosine functions might equal each other. 1. **Using the identity**: We can also set: \[ \sin(\theta - 20^{\circ}) = \frac{1}{8.5} \cos(3\theta) \] This will require numerical or graphical methods to find additional solutions. ### Final General Solution The general solution we have found so far is: \[ \theta = 20^{\circ} + k \cdot 360^{\circ} \quad \text{for any integer } k \] To find additional solutions, we would typically use numerical methods or graphing techniques to find intersections of the two functions. If you would like to explore numerical solutions or specific values further, please let me know!