Find the values of `theta ` between `0^(@) and 360^(@)` which satisfy the equation `3 cos 2 theta - sin theta = 2.`

Answer

To solve the equation \(3 \cos 2\theta - \sin \theta = 2\) for values of \(\theta\) between \(0^\circ\) and \(360^\circ\), we can follow these steps:

### Step 1: Rewrite the equation using the double angle identity
We know that \(\cos 2\theta = 1 - 2\sin^2 \theta\). Therefore, we can substitute this into the equation:

\[ 3(1 - 2\sin^2 \theta) - \sin \theta = 2 \]

### Step 2: Simplify the equation
Expanding the equation gives:

\[ 3 - 6\sin^2 \theta - \sin \theta = 2 \]

Now, rearranging the equation leads to:

\[ -6\sin^2 \theta - \sin \theta + 1 = 0 \]

### Step 3: Multiply through by -1
To make the equation easier to work with, we multiply the entire equation by -1:

\[ 6\sin^2 \theta + \sin \theta - 1 = 0 \]

### Step 4: Factor the quadratic equation
Next, we need to factor the quadratic equation. We can look for two numbers that multiply to \(6 \times -1 = -6\) and add to \(1\). The numbers \(3\) and \(-2\) work:

\[ 6\sin^2 \theta + 3\sin \theta - 2\sin \theta - 1 = 0 \]

Grouping the terms:

\[ (6\sin^2 \theta + 3\sin \theta) + (-2\sin \theta - 1) = 0 \]

Factoring by grouping:

\[ 3\sin \theta(2\sin \theta + 1) - 1(2\sin \theta + 1) = 0 \]

Factoring out the common factor:

\[ (3\sin \theta - 1)(2\sin \theta + 1) = 0 \]

### Step 5: Solve for \(\sin \theta\)
Setting each factor to zero gives us two equations:

1. \(3\sin \theta - 1 = 0\)
2. \(2\sin \theta + 1 = 0\)

From the first equation:

\[ 3\sin \theta = 1 \implies \sin \theta = \frac{1}{3} \]

From the second equation:

\[ 2\sin \theta = -1 \implies \sin \theta = -\frac{1}{2} \]

### Step 6: Find the angles \(\theta\)
1. For \(\sin \theta = \frac{1}{3}\):
- \(\theta = \sin^{-1}\left(\frac{1}{3}\right)\)
- The second solution in the range \(0^\circ\) to \(360^\circ\) is \(\theta = 180^\circ - \sin^{-1}\left(\frac{1}{3}\right)\).

2. For \(\sin \theta = -\frac{1}{2}\):
- The solutions are \(\theta = 210^\circ\) and \(\theta = 330^\circ\).

### Step 7: Compile all solutions
Thus, the values of \(\theta\) that satisfy the equation \(3 \cos 2\theta - \sin \theta = 2\) in the interval \(0^\circ\) to \(360^\circ\) are:

\[ \theta = \sin^{-1}\left(\frac{1}{3}\right), \quad 180^\circ - \sin^{-1}\left(\frac{1}{3}\right), \quad 210^\circ, \quad 330^\circ \]