The number of values of `theta` in `[0, 2pi]` that satisfies the equation `sin^(2)theta - cos theta = (1)/(4)`

Options

A. 4
B. 3
C. 2
D. None of these

Answer

To solve the equation \( \sin^2 \theta - \cos \theta = \frac{1}{4} \) for the number of values of \( \theta \) in the interval \( [0, 2\pi] \), we can follow these steps:

### Step 1: Rewrite the equation using the Pythagorean identity
We know that \( \sin^2 \theta + \cos^2 \theta = 1 \). Thus, we can express \( \sin^2 \theta \) in terms of \( \cos \theta \):
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Substituting this into the original equation gives:
\[ 1 - \cos^2 \theta - \cos \theta = \frac{1}{4} \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation:
\[ 1 - \cos^2 \theta - \cos \theta - \frac{1}{4} = 0 \]
This simplifies to:
\[ 1 - \frac{1}{4} - \cos^2 \theta - \cos \theta = 0 \]
\[ \frac{3}{4} - \cos^2 \theta - \cos \theta = 0 \]
Multiplying through by 4 to eliminate the fraction:
\[ 3 - 4\cos^2 \theta - 4\cos \theta = 0 \]

### Step 3: Rearranging into standard quadratic form
Rearranging gives us a standard quadratic equation:
\[ 4\cos^2 \theta + 4\cos \theta - 3 = 0 \]

### Step 4: Factor the quadratic equation
We can factor this quadratic equation:
\[ (4\cos \theta - 3)(\cos \theta + 1) = 0 \]
This gives us two equations to solve:
1. \( 4\cos \theta - 3 = 0 \)
2. \( \cos \theta + 1 = 0 \)

### Step 5: Solve for \( \cos \theta \)
From the first equation:
\[ 4\cos \theta = 3 \implies \cos \theta = \frac{3}{4} \]
From the second equation:
\[ \cos \theta = -1 \]

### Step 6: Find the values of \( \theta \)
1. For \( \cos \theta = \frac{3}{4} \):
- The values of \( \theta \) in the interval \( [0, 2\pi] \) are:
\[ \theta = \cos^{-1}\left(\frac{3}{4}\right) \quad \text{and} \quad \theta = 2\pi - \cos^{-1}\left(\frac{3}{4}\right) \]
- This gives us two solutions.

2. For \( \cos \theta = -1 \):
- The value of \( \theta \) is:
\[ \theta = \pi \]
- This gives us one solution.

### Step 7: Count the total number of solutions
Adding the solutions together, we have:
- 2 solutions from \( \cos \theta = \frac{3}{4} \)
- 1 solution from \( \cos \theta = -1 \)

Thus, the total number of values of \( \theta \) that satisfy the equation in the interval \( [0, 2\pi] \) is:
\[ \text{Total solutions} = 2 + 1 = 3 \]

### Final Answer
The number of values of \( \theta \) in \( [0, 2\pi] \) that satisfies the equation is **3**.