If `sin theta=(sqrt(3))/(2)" then " ("cosec"theta+cot theta)=?`

Options

A. `(2+sqrt(3))`
B. `2sqrt(3)`
C. `sqrt(2)`
D. `sqrt(3)`

Answer

To solve the problem where \( \sin \theta = \frac{\sqrt{3}}{2} \) and we need to find \( \csc \theta + \cot \theta \), we will follow these steps:

### Step 1: Identify the values of sine, cosecant, and cotangent
Given that \( \sin \theta = \frac{\sqrt{3}}{2} \), we can find \( \csc \theta \) and \( \cot \theta \).

### Step 2: Calculate cosecant
The cosecant function is the reciprocal of the sine function:
\[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 3: Construct a right triangle
To find \( \cot \theta \), we can construct a right triangle where:
- The opposite side (perpendicular) is \( \sqrt{3} \)
- The hypotenuse is \( 2 \)

Using the Pythagorean theorem, we can find the adjacent side (base):
\[ \text{Adjacent}^2 + \text{Opposite}^2 = \text{Hypotenuse}^2 \]
\[ \text{Adjacent}^2 + (\sqrt{3})^2 = 2^2 \]
\[ \text{Adjacent}^2 + 3 = 4 \]
\[ \text{Adjacent}^2 = 1 \implies \text{Adjacent} = 1 \]

### Step 4: Calculate cotangent
The cotangent function is the ratio of the adjacent side to the opposite side:
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\sqrt{3}} \]

### Step 5: Add cosecant and cotangent
Now we can find \( \csc \theta + \cot \theta \):
\[ \csc \theta + \cot \theta = \frac{2}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{2 + 1}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

### Final Answer
Thus, the value of \( \csc \theta + \cot \theta \) is:
\[ \sqrt{3} \]