The minimum and maximum values of expression `y=costheta(sintheta+sqrt(sin^2theta+sin^2alpha))`

Options

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

Answer

To find the minimum and maximum values of the expression \( y = \cos \theta \left( \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \cos \theta \left( \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right) \]

### Step 2: Isolate \( \sec \theta \)
We can rewrite \( \cos \theta \) as \( \frac{1}{\sec \theta} \):
\[ y = \frac{1}{\sec \theta} \left( \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \right) \]
Multiplying both sides by \( \sec \theta \):
\[ y \sec \theta = \sin \theta + \sqrt{\sin^2 \theta + \sin^2 \alpha} \]

### Step 3: Rearranging the equation
Rearranging gives:
\[ y \sec \theta - \sin \theta = \sqrt{\sin^2 \theta + \sin^2 \alpha} \]

### Step 4: Square both sides
Squaring both sides to eliminate the square root:
\[ (y \sec \theta - \sin \theta)^2 = \sin^2 \theta + \sin^2 \alpha \]

### Step 5: Expand the left side
Expanding the left side:
\[ y^2 \sec^2 \theta - 2y \sin \theta \sec \theta + \sin^2 \theta = \sin^2 \theta + \sin^2 \alpha \]
Cancelling \( \sin^2 \theta \) from both sides:
\[ y^2 \sec^2 \theta - 2y \sin \theta \sec \theta = \sin^2 \alpha \]

### Step 6: Substitute \( \tan \theta \)
Let \( x = \tan \theta \), then \( \sec^2 \theta = 1 + x^2 \):
\[ y^2 (1 + x^2) - 2y x = \sin^2 \alpha \]
This can be rearranged to form a quadratic equation in \( x \):
\[ y^2 x^2 - 2y x + (y^2 - \sin^2 \alpha) = 0 \]

### Step 7: Find the discriminant
For \( x \) to be real, the discriminant must be non-negative:
\[ D = (-2y)^2 - 4y^2(y^2 - \sin^2 \alpha) \geq 0 \]
This simplifies to:
\[ 4y^2 - 4y^2(y^2 - \sin^2 \alpha) \geq 0 \]
Factoring out \( 4y^2 \):
\[ 4y^2(1 - y^2 + \sin^2 \alpha) \geq 0 \]

### Step 8: Solve the inequality
Since \( 4y^2 \geq 0 \), we need:
\[ 1 - y^2 + \sin^2 \alpha \geq 0 \]
This leads to:
\[ y^2 \leq 1 + \sin^2 \alpha \]

### Step 9: Determine the bounds for \( y \)
Taking square roots gives:
\[ |y| \leq \sqrt{1 + \sin^2 \alpha} \]
Thus, the maximum and minimum values of \( y \) are:
\[ -\sqrt{1 + \sin^2 \alpha} \leq y \leq \sqrt{1 + \sin^2 \alpha} \]

### Conclusion
The minimum value of \( y \) is \( -\sqrt{1 + \sin^2 \alpha} \) and the maximum value is \( \sqrt{1 + \sin^2 \alpha} \).

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