If the equation `sin^(2)theta-costheta=(1)/(4),` then the value of `theta` lying in the interval `0lethetale2pi` is

Options

A. `(pi)/(3),(5pi)/(3)`
B. `(pi)/(3),(2pi)/(3)`
C. `(4pi)/(3),(5pi)/(3)`
D. `(3pi)/(5),(pi)/(5)`

Explanation

To solve the equation \( \sin^2 \theta - \cos \theta = \frac{1}{4} \), we will follow these steps:

### Step 1: Rewrite the equation
We know that \( \sin^2 \theta = 1 - \cos^2 \theta \). We can substitute this into the equation:

\[ 1 - \cos^2 \theta - \cos \theta = \frac{1}{4} \]

### Step 2: Rearrange the equation
Now, we rearrange the equation to bring all terms to one side:

\[ 1 - \frac{1}{4} - \cos^2 \theta - \cos \theta = 0 \]

Calculating \( 1 - \frac{1}{4} \):

\[ \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

So the equation becomes:

\[ -\cos^2 \theta - \cos \theta + \frac{3}{4} = 0 \]

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

\[ \cos^2 \theta + \cos \theta - \frac{3}{4} = 0 \]

### Step 4: Multiply through by 4
To eliminate the fraction, we can multiply the entire equation by 4:

\[ 4\cos^2 \theta + 4\cos \theta - 3 = 0 \]

### Step 5: Use the quadratic formula
This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 4, b = 4, c = -3 \). We can apply the quadratic formula:

\[ \cos \theta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values:

\[ \cos \theta = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 4 \cdot (-3)}}{2 \cdot 4} \]

Calculating the discriminant:

\[ \cos \theta = \frac{-4 \pm \sqrt{16 + 48}}{8} \]
\[ \cos \theta = \frac{-4 \pm \sqrt{64}}{8} \]
\[ \cos \theta = \frac{-4 \pm 8}{8} \]

### Step 6: Solve for \( \cos \theta \)
This gives us two potential solutions:

1. \( \cos \theta = \frac{4}{8} = \frac{1}{2} \)
2. \( \cos \theta = \frac{-12}{8} = -\frac{3}{2} \) (not valid since the range of cosine is \([-1, 1]\))

So we only consider \( \cos \theta = \frac{1}{2} \).

### Step 7: Find the angles
The angles for which \( \cos \theta = \frac{1}{2} \) in the interval \( [0, 2\pi] \) are:

\[ \theta = \frac{\pi}{3}, \quad \text{and} \quad \theta = \frac{5\pi}{3} \]

### Final Answer
Thus, the values of \( \theta \) lying in the interval \( [0, 2\pi] \) are:

\[ \theta = \frac{\pi}{3}, \quad \frac{5\pi}{3} \]

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