QUESTION 1 (10 marks) Solve \( 3 \cos 2 \theta+0.6=0 \) for \( 0 \leq \theta<2 \pi \)

To solve the equation \( 3 \cos(2\theta) + 0.6 = 0 \) for \( 0 \leq \theta < 2\pi \), follow these steps: ### Step 1: Isolate the Trigonometric Function Start by isolating the cosine term: \[ 3 \cos(2\theta) + 0.6 = 0 \] Subtract 0.6 from both sides: \[ 3 \cos(2\theta) = -0.6 \] Divide both sides by 3: \[ \cos(2\theta) = -0.2 \] ### Step 2: Find the General Solutions for \( 2\theta \) The equation \( \cos(2\theta) = -0.2 \) can be solved by finding the general solutions for \( 2\theta \): \[ 2\theta = \cos^{-1}(-0.2) + 2k\pi \quad \text{and} \quad 2\theta = -\cos^{-1}(-0.2) + 2k\pi \quad \text{for integer } k \] Calculate \( \cos^{-1}(-0.2) \): \[ \cos^{-1}(-0.2) \approx 1.772 \text{ radians} \] Thus, the general solutions for \( 2\theta \) are: \[ 2\theta \approx 1.772 + 2k\pi \quad \text{and} \quad 2\theta \approx -1.772 + 2k\pi \] ### Step 3: Solve for \( \theta \) Within the Specified Interval Divide both sides by 2 to solve for \( \theta \): \[ \theta \approx \frac{1.772}{2} + k\pi \quad \text{and} \quad \theta \approx \frac{-1.772}{2} + k\pi \] Simplify: \[ \theta \approx 0.886 + k\pi \quad \text{and} \quad \theta \approx -0.886 + k\pi \] Now, find all solutions within \( 0 \leq \theta < 2\pi \): 1. For \( k = 0 \): - \( \theta \approx 0.886 \) radians - \( \theta \approx -0.886 + \pi \approx 2.255 \) radians 2. For \( k = 1 \): - \( \theta \approx 0.886 + \pi \approx 4.027 \) radians - \( \theta \approx -0.886 + 2\pi \approx 5.397 \) radians ### Step 4: List All Solutions The solutions within the interval \( 0 \leq \theta < 2\pi \) are: \[ \theta \approx 0.886 \text{ radians},\ 2.256 \text{ radians},\ 4.027 \text{ radians},\ 5.397 \text{ radians} \] ### Final Answer The solutions for \( \theta \) in the interval \( 0 \leq \theta < 2\pi \) are approximately: \[ \theta \approx 0.886,\ 2.256,\ 4.027,\ \text{and } 5.397 \text{ radians} \]