If `"cosec"theta=2`,then find the value of `"cosec" 3theta`

Answer

To solve the problem where we need to find the value of \( \csc 3\theta \) given that \( \csc \theta = 2 \), we can follow these steps:

### Step 1: Understand the given information
We know that \( \csc \theta = 2 \). The cosecant function is the reciprocal of the sine function, so we can express this as:
\[ \csc \theta = \frac{1}{\sin \theta} \]
Thus, we can write:
\[ \sin \theta = \frac{1}{\csc \theta} = \frac{1}{2} \]

### Step 2: Find the angle \( \theta \)
To find \( \theta \), we can use the inverse sine function:
\[ \theta = \sin^{-1}\left(\frac{1}{2}\right) \]
From trigonometric values, we know that:
\[ \sin 30^\circ = \frac{1}{2} \]
Thus, we have:
\[ \theta = 30^\circ \]

### Step 3: Calculate \( 3\theta \)
Now, we need to find \( 3\theta \):
\[ 3\theta = 3 \times 30^\circ = 90^\circ \]

### Step 4: Find \( \csc 3\theta \)
Now we can find \( \csc 3\theta \):
\[ \csc 3\theta = \csc 90^\circ \]
From trigonometric values, we know that:
\[ \csc 90^\circ = 1 \]

### Final Answer
Thus, the value of \( \csc 3\theta \) is:
\[ \csc 3\theta = 1 \]