The smallest positive angle (in degree) which satisfies the equation `2sin^2theta+sqrt(3)costheta+1=0` is

Answer

To solve the equation \( 2\sin^2\theta + \sqrt{3}\cos\theta + 1 = 0 \), we will follow these steps:

### Step 1: Rewrite the equation using the Pythagorean identity
We know that \( \sin^2\theta = 1 - \cos^2\theta \). Therefore, we can substitute \( \sin^2\theta \) in the equation:

\[ 2(1 - \cos^2\theta) + \sqrt{3}\cos\theta + 1 = 0 \]

### Step 2: Simplify the equation
Expanding the equation gives:

\[ 2 - 2\cos^2\theta + \sqrt{3}\cos\theta + 1 = 0 \]

Combining like terms results in:

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

### Step 3: Multiply through by -1
To make the leading coefficient positive, we multiply the entire equation by -1:

\[ 2\cos^2\theta - \sqrt{3}\cos\theta - 3 = 0 \]

### Step 4: Use the quadratic formula
This is a quadratic equation in the form \( A\cos^2\theta + B\cos\theta + C = 0 \), where \( A = 2 \), \( B = -\sqrt{3} \), and \( C = -3 \). We can apply the quadratic formula:

\[ \cos\theta = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting the values:

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

### Step 5: Calculate the discriminant
Calculating the discriminant:

\[ (\sqrt{3})^2 - 4 \cdot 2 \cdot (-3) = 3 + 24 = 27 \]

### Step 6: Substitute back into the formula
Now substituting back into the formula:

\[ \cos\theta = \frac{\sqrt{3} \pm \sqrt{27}}{4} = \frac{\sqrt{3} \pm 3\sqrt{3}}{4} \]

### Step 7: Simplify the results
This gives us two cases:

1. \( \cos\theta = \frac{4\sqrt{3}}{4} = \sqrt{3} \) (not possible since \( \cos\theta \) cannot exceed 1)
2. \( \cos\theta = \frac{-2\sqrt{3}}{4} = -\frac{\sqrt{3}}{2} \)

### Step 8: Determine the angle for \( \cos\theta = -\frac{\sqrt{3}}{2} \)
The cosine value of \( -\frac{\sqrt{3}}{2} \) corresponds to angles in the second and third quadrants. The reference angle is \( \frac{\pi}{6} \). Thus:

\[ \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \quad \text{(in the second quadrant)} \]

### Step 9: Convert to degrees
To find the smallest positive angle in degrees:

\[ \theta = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ \]

### Final Answer
The smallest positive angle \( \theta \) that satisfies the equation is:

\[ \boxed{150^\circ} \]