General value of `theta` obtained from the equation `cos 2 theta = sin alpha` is

Options

A. `2 theta = (pi//2) - alpha`
B. `theta = 2 n pi +- (pi//2 - alpha)`
C. `theta = 1/2[n pi + (-1)^(n) cdot alpha]`
D. `theta = n pi +- ((pi)/4 - alpha/2)`

Answer

To find the general value of \( \theta \) from the equation \( \cos 2\theta = \sin \alpha \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \cos 2\theta = \sin \alpha \]

### Step 2: Use the Co-Function Identity
We know that:
\[ \sin \alpha = \cos\left(90^\circ - \alpha\) \]
Thus, we can rewrite our equation as:
\[ \cos 2\theta = \cos\left(90^\circ - \alpha\right) \]

### Step 3: Apply the Cosine Identity
From the property of cosine, if \( \cos A = \cos B \), then:
\[ A = 2n\pi \pm B \]
Applying this to our equation gives:
\[ 2\theta = 2n\pi \pm (90^\circ - \alpha) \]

### Step 4: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = n\pi \pm \frac{90^\circ - \alpha}{2} \]

### Step 5: Simplify the Expression
We can express \( 90^\circ \) in radians as \( \frac{\pi}{2} \):
\[ \theta = n\pi \pm \left(\frac{\pi}{4} - \frac{\alpha}{2}\right) \]

### Final General Value of \( \theta \)
Thus, the general value of \( \theta \) can be written as:
\[ \theta = n\pi \pm \frac{\pi}{4} - \frac{\alpha}{2} \]