If `abs(cos theta{sin theta+sqrt(sin^2theta+sin^2alpha)})lek`, then the value of k

Options

A. `sqrt(1+cos^2alpha)`
B. `sqrt(1+sin^2alpha)`
C. `sqrt(2+sin^2alpha)`
D. `sqrt(2+cos^2alpha)`

Explanation

To solve the problem, we need to find the value of \( k \) such that

\[ \left| \cos \theta \left( \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right) \right| \leq k. \]

Let's denote the expression inside the absolute value as \( U \):

\[ U = \cos \theta \left( \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right). \]

### Step 1: Rewrite the expression

We can rewrite \( U \) as:

\[ U = \cos \theta \sin \theta + \cos \theta \sqrt{\sin^2 \theta + \sin^2 \alpha}. \]

### Step 2: Analyze the components

We will analyze the two components of \( U \):

1. The first term is \( \cos \theta \sin \theta \).
2. The second term is \( \cos \theta \sqrt{\sin^2 \theta + \sin^2 \alpha} \).

### Step 3: Find the maximum value of \( U \)

To find the maximum value of \( U \), we can consider the expression:

\[ U = \cos \theta \sin \theta + \cos \theta \sqrt{\sin^2 \theta + \sin^2 \alpha}. \]

### Step 4: Use trigonometric identities

Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can express \( \cos \theta \sin \theta \) as:

\[ \cos \theta \sin \theta = \frac{1}{2} \sin 2\theta. \]

### Step 5: Square both sides

To find the maximum value, we can square both sides:

\[ U^2 = \left( \cos \theta \sin \theta + \cos \theta \sqrt{\sin^2 \theta + \sin^2 \alpha} \right)^2. \]

### Step 6: Expand the squared expression

Expanding the squared expression gives:

\[ U^2 = \cos^2 \theta \sin^2 \theta + 2 \cos^2 \theta \sin \theta \sqrt{\sin^2 \theta + \sin^2 \alpha} + \cos^2 \theta (\sin^2 \theta + \sin^2 \alpha). \]

### Step 7: Set up the discriminant condition

To ensure that \( U \) is real, we need to set up the discriminant condition for the quadratic in terms of \( \tan \theta \):

\[ B^2 - 4AC \geq 0, \]

where \( A = U^2 \), \( B = -2U \), and \( C = U^2 - \sin^2 \alpha \).

### Step 8: Solve the discriminant inequality

The discriminant simplifies to:

\[ 4U^2 - 4(U^2 - \sin^2 \alpha) \geq 0. \]

This leads to:

\[ 4\sin^2 \alpha \geq 0, \]

which is always true.

### Step 9: Find the maximum value of \( U \)

From the analysis, we find that:

\[ U^2 \leq 1 + \sin^2 \alpha. \]

Thus, taking the square root gives:

\[ |U| \leq \sqrt{1 + \sin^2 \alpha}. \]

### Conclusion

The maximum value of \( k \) is:

\[ k = \sqrt{1 + \sin^2 \alpha}. \]