For which acute angle `theta,"cosec"^(2)theta=3sqrt(3)cottheta-5`?

Options

A. `(5pi)/(12)`
B. `(pi)/(3)`
C. `(pi)/(6)`
D. `(pi)/(4)`

Explanation

To solve the equation \( \csc^2 \theta = 3\sqrt{3} \cot \theta - 5 \), we will follow these steps:

### Step 1: Rewrite the equation using trigonometric identities
We know that \( \csc^2 \theta = 1 + \cot^2 \theta \). Therefore, we can rewrite the equation as:
\[ 1 + \cot^2 \theta = 3\sqrt{3} \cot \theta - 5 \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ \cot^2 \theta - 3\sqrt{3} \cot \theta + 1 + 5 = 0 \]
This simplifies to:
\[ \cot^2 \theta - 3\sqrt{3} \cot \theta + 6 = 0 \]

### Step 3: Identify coefficients for the quadratic formula
In the quadratic equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -3\sqrt{3} \)
- \( c = 6 \)

### Step 4: Apply the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), we substitute the values:
\[ \cot \theta = \frac{-(-3\sqrt{3}) \pm \sqrt{(-3\sqrt{3})^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1} \]
This simplifies to:
\[ \cot \theta = \frac{3\sqrt{3} \pm \sqrt{27 - 24}}{2} \]
\[ \cot \theta = \frac{3\sqrt{3} \pm \sqrt{3}}{2} \]

### Step 5: Simplify the expression
Factoring out \( \sqrt{3} \):
\[ \cot \theta = \frac{\sqrt{3}(3 \pm 1)}{2} \]
This gives us two possible values:
1. \( \cot \theta = \frac{4\sqrt{3}}{2} = 2\sqrt{3} \)
2. \( \cot \theta = \frac{2\sqrt{3}}{2} = \sqrt{3} \)

### Step 6: Find the corresponding angles
1. For \( \cot \theta = \sqrt{3} \):
- This corresponds to \( \theta = 30^\circ \) or \( \theta = \frac{\pi}{6} \) radians.

2. For \( \cot \theta = 2\sqrt{3} \):
- This does not correspond to an acute angle since \( \cot \theta \) must be positive and less than \( \infty \).

### Conclusion
The acute angle \( \theta \) that satisfies the equation is:
\[ \theta = 30^\circ \text{ or } \theta = \frac{\pi}{6} \]

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