If ` 2 cos theta + 2 sqrt(2) = 3 sec theta ` where ` theta in (0, 2 pi) ` then which of the following can be correct ?

Options

A. `cos theta= (1)/(sqrt(2))`
B. `tan theta=1`
C. `sin theta = - (1)/(sqrt(2))`
D. `cot theta = - 1`

Answer

To solve the equation \( 2 \cos \theta + 2 \sqrt{2} = 3 \sec \theta \) where \( \theta \) is in the interval \( (0, 2\pi) \), we will follow these steps:

### Step 1: Rewrite the equation in terms of cosine
We know that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore, we can rewrite the equation as:
\[ 2 \cos \theta + 2 \sqrt{2} = \frac{3}{\cos \theta} \]

### Step 2: Multiply through by \( \cos \theta \)
To eliminate the fraction, we multiply both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)):
\[ 2 \cos^2 \theta + 2 \sqrt{2} \cos \theta = 3 \]

### Step 3: Rearrange the equation
Rearranging gives us a standard quadratic form:
\[ 2 \cos^2 \theta + 2 \sqrt{2} \cos \theta - 3 = 0 \]

### Step 4: Identify coefficients
Let \( x = \cos \theta \). The equation becomes:
\[ 2x^2 + 2\sqrt{2}x - 3 = 0 \]
Here, \( a = 2 \), \( b = 2\sqrt{2} \), and \( c = -3 \).

### Step 5: Use the quadratic formula
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-2\sqrt{2} \pm \sqrt{(2\sqrt{2})^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ (2\sqrt{2})^2 = 8 \quad \text{and} \quad 4 \cdot 2 \cdot (-3) = -24 \]
Thus:
\[ b^2 - 4ac = 8 + 24 = 32 \]

### Step 6: Solve for \( x \)
Now substituting back into the formula:
\[ x = \frac{-2\sqrt{2} \pm \sqrt{32}}{4} \]
Since \( \sqrt{32} = 4\sqrt{2} \):
\[ x = \frac{-2\sqrt{2} \pm 4\sqrt{2}}{4} \]
This simplifies to:
\[ x = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2} \quad \text{or} \quad x = \frac{-6\sqrt{2}}{4} = -\frac{3\sqrt{2}}{2} \]

### Step 7: Analyze the solutions
Since \( \cos \theta \) must be in the range \([-1, 1]\), we discard \( -\frac{3\sqrt{2}}{2} \) as it is not valid. Thus, we have:
\[ \cos \theta = \frac{\sqrt{2}}{2} \]

### Step 8: Find the angles
The angles corresponding to \( \cos \theta = \frac{\sqrt{2}}{2} \) in the interval \( (0, 2\pi) \) are:
\[ \theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{7\pi}{4} \]

### Conclusion
Thus, the values of \( \theta \) that satisfy the equation are \( \frac{\pi}{4} \) and \( \frac{7\pi}{4} \).

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